Probability Seminars at Cornell

Welcome to the experimental and yet unnoficial page for the Probability Seminars of the Department of Mathematics at Cornell. The current organizers of the seminar are Ahmed El Alaoui, Lionel Levine, and Victor Souza.

2025 — 26 To be announced... Kengo Kato (Cornell University) Monday, 1st of December 2025 at in Mallot 406 Probability To be announced... Rohan Sarkar (Binghamton University) Monday, 24th of November 2025 at in Malott 406 Probability To be announced... Rohit Lamba (Cornell University) Monday, 17th of November 2025 at in Mallot 406 Probability Fundamental Limits of Low-Rank Matrix Estimation: Information-Theoretic and Computational Perspectives Yuchen Wu (Cornell University) Monday, 10th of November 2025 at in Mallot 406 Probability Many statistical estimation problems can be reduced to the reconstruction of a low-rank $n \times d$ matrix when observed through a noisy channel. While tremendous positive results have been established, relatively few works focus on understanding the fundamental limitations of the proposed models and algorithms. Understanding such limitations not only provides practitioners with guidance on algorithm selection, but also spurs the development of cutting-edge methodologies. In this talk, I will present some recent progress in this direction from two perspectives in the context of low-rank matrix estimation. From an information-theoretic perspective, I will give an exact characterization of the limiting minimum estimation error. Our results apply to the high-dimensional regime $n,d \to \infty$ and $d/n\to \infty$ (or $d/n \to 0$) and generalize earlier works that focus on the proportional asymptotics $n,d \to \infty$, $d/n\to \delta \in (0,\infty)$. From an algorithmic perspective, large-dimensional matrices are often processed by iterative algorithms like power iteration and gradient descent, thus encouraging the pursuit of understanding the fundamental limits of these approaches. We introduce a class of general first order methods (GFOM), which is broad enough to include the aforementioned algorithms and many others. I will describe the asymptotic behavior of any GFOM, and provide a sharp characterization of the optimal error achieved by the GFOM class. This is based on joint works with Michael Celentano and Andrea Montanari. Fast relaxation of the random field Ising dynamics Ahmed El Alaoui (Cornell University) Monday, 3rd of November 2025 at in Mallot 406 Probability I will present new results on Glauber dynamics for the ferromagnetic Random Field Ising Model (RFIM) in $\mathbb{Z}^d$. Of particular interest is a region of the phase diagram called the Griffiths phase where correlations decay exponentially fast on average over the random field, but there exists arbitrarily large islands of weak fields where low temperature behavior takes place, and quenched correlations do not decay. We prove a weak Poincaré inequality, implying that the dynamics relax in polynomial time in the volume of the domain and that efficient samplers exist in this regime. The proof combines ideas from stochastic analysis with coarse graining techniques. This is based on joint work with Ronen Eldan, Reza Gheissari, and Arianna Piana. Pseudo-Maximum Likelihood Theory for High-Dimensional Rank-One Inference Justin Ko (Syracuse University) Monday, 27th of October 2025 at in Mallot 406 Probability We consider the task of estimating a rank-one matrix from noisy observations. Models that fall in this framework include community detection and spiked Wigner models. In this talk, I will discuss pseudo-maximum likelihood theory for such inference problems. We provide a variational formula for the asymptotic maximum pseudo-likelihood and characterize the asymptotic performance of pseudo maximum likelihood estimators. We will also discuss the implications of these findings to least squares estimators. Our approach uses the recent connections between statistical inference, statistical physics and random matrix theory, and in particular the connection between the maximum likelihood and the ground state of a modified spin glass. This is based on joint work with Curtis Grant and Aukosh Jagannath. The logic of floor division Ben Przybocki (Carnegie Mellon University) Monday, 20th of October 2025 at in Malott 406 Probability Floor division (also called integer division or Euclidean division) is the operation that divides two integers and discards the remainder. Floor division satisfies some nontrivial identities, such as this one proposed by Ramanujan in 1918: $$\Big\lfloor\frac{n}{3}\Big\rfloor + \Big\lfloor\frac{n+2}{6}\Big\rfloor + \Big\lfloor\frac{n+4}{6}\Big\rfloor = \Big\lfloor\frac{n}{2}\Big\rfloor + \Big\lfloor\frac{n+3}{6}\Big\rfloor.$$ How can we systematically verify such an equation (without resorting to case bashing)? And can we characterize the set of all such equations in a perspicuous way? To answer these questions, I present a sound and complete axiomatization for the equational theory of linear arithmetic with operations for floor division by fixed positive integers. My axiomatization has five axiom schemata. Three of these are somewhat obvious, but the remaining two are more interesting. One is a variant of Hermite's identity, discovered in 1884. The other appears to be a hitherto unnoticed identity, which is perhaps surprising for an identity involving only elementary operations. I prove that my axiomatization is complete by showing that the axioms suffice to put every term into a normal form, and this normal form may be of independent interest. My hope is that, by the end of this talk, you will have the tools to develop a greater dexterity in manipulating expressions involving floor division, so that you can readily see the truth of identities like the one above. 2024 — 25 Continuity of asymptotic entropy on groups Eduardo Silva (University of Münster) Monday, 5th of May 2025 at in Malott 406 Probability The asymptotic entropy of a random walk on a countable group is a non-negative number that determines the existence of non-constant bounded harmonic functions on the group. A natural question to ask is whether the asymptotic entropy, seen as a function of the step distribution of the random walk, is continuous. In this talk, I will explain two recent results on the continuity of asymptotic entropy: one for groups whose Poisson boundaries can be identified with a compact metric space carrying a unique stationary measure, and another for wreath products $A \wr \mathbb{Z}^d$, where $A$ is a countable group and $d \geq 3$. Existence of Full Replica Symmetry Breaking for the Sherrington-Kirkpatrick Model Yuxin Zhou (University of Chicago) Monday, 21st of April 2025 at in Malott 406 Probability The Sherrington–Kirkpatrick (SK) model is a well-known model for spin glasses originated in physics. In a series of ground-breaking articles by Giorgio Parisi proposed in 1979, it was predicted that the limiting free energy of the SK model is given by a variational principle, known as the Parisi formula and its minimizer, known as the Parisi measure will be full replica symmetry breaking (FRSB) at low temperature. The former one was confirmed in the past decade. However the existence of FRSB has not been established before.

In this talk, I will show that the Parisi measure of the SK model is FRSB slightly beyond the high temperature. I will also talk about the structure of the corresponding Parisi measure. Ergodicity of the stochastic nonlinear Schrodinger equation Hung Nguyen (University of Tennessee) Monday, 14th of April 2025 at in Malott 406 Probability We discuss the long-time behaviors of the nonlinear Schrodinger equation while being subjected to a random perturbation via Gaussian additive noise. It is well-known that, under the impact of sufficiently strong damping, the dynamics possesses a unique invariant probability measure. In this talk, we address the mixing issue and demonstrate that the solutions are attracted toward equilibrium with polynomial rates of any order. Our approach relies on a coupling strategy making use of pathwise Strichartz estimates specifically derived for the equation. This talk is based on a joint work with Kihoon Seong. A remarkable example of stochastic processes with long-range dependence Yizao Wang (University of Cincinnati) Monday, 7th of April 2025 at in Malott 406 Probability We consider a class of regularly-varying stochastic processes with long-range dependence introduced first by Rosinski and Samorodnitsky in 1996. The original model can be represented as stochastic integrals with respect to a symmetric alpha-stable random measure, and we investigate an extension of the model that can be represented as multiple-stochastic integrals. When the multiplicity p is greater than or equal to 2 a new phase transition occurs. In the sub-critical regime, we reveal a delicate asymptotic behavior of extremes: the so-called candidate extremal index and the extremal index are not the same. We are not aware of any other regularly varying processes with such a feature.

Based on joint works with Shuyang Bai and Rafal Kulik. Homological percolation in a torus Matt Kahle (Ohio State University) Monday, 24th of March 2025 at in Malott 406 Probability The celebrated Harris–Kesten theorem is that the critical probability for bond percolation in the square lattice is $1/2$. It has been a folklore problem for apparently some time to find an appropriate analogue of this theorem in high dimensions.

Duncan, Schweinhart, and I studied "plaquette percolation" in a $d$-dimensional torus made by identifying opposite sides of a subdivided $d$-dimensional cube, where $k$-dimensional cubical cells are inserted independently with probability $p$. For an appropriate homological definition of "giant cycles", and whenever $d = 2k$, we show that the critical probability is indeed $1/2$.

Since this is a talk in probability seminar, I will aim to focus on the topological definitions and main ideas. The talk will be mostly self-contained. Models of random spanning trees Moon Duchin (Cornell University) Monday, 17th of March 2025 at in Malott 406 Probability Suppose you want a random spanning tree in a given graph $G$. The two most popular ways to do this are UST (uniformly random) and MST (minimum, by random weights).  They are not the same, but there is surprisingly little literature on how the probability distributions differ!  I'll give some results on the quantitative properties of MST that justify the slogan "Greedy trees are bushy."  This will end up delivering some new insight into so-called intransitive dice, where you can have three random variables so that $A>B$, $B>C$, and $C>A$ are all more likely than not. (Based on joint work with Eric Babson, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, and Jamie Tucker-Foltz.) Universality for Self-Organized Criticality Hyojeong Son (University of Washington) Monday, 10th of March 2025 at in Malott 406 Probability Self-organized criticality (SOC), introduced by Bak, Tang, and Wiesenfeld in 1987, describes how complex systems naturally evolve to a critical state without the need for fine-tuned parameters. SOC is observed in diverse phenomena, including financial market fluctuations and forest fires, which exhibit power-law distributions. While traditional models like sandpiles have been instrumental in studying SOC, the Activated Random Walk (ARW) model stands out as a promising candidate due to its universal characteristics. In this talk, we review existing SOC models and present evidence supporting the ARW model’s robustness and versatility in capturing self-organized criticality. This is joint work with Madeline Brown and Christopher Hoffman. Heat Kernels of Schrödinger Operators on Manifolds Anthony  Graves-McCleary (Cornell University) Monday, 3rd of March 2025 at in Malott 406 Probability We present heat kernel estimates for Schrödinger operators $\Delta + W$ on weighted Riemannian manifolds. We study manifolds on which Brownian motion is recurrent; this offers some interesting technical obstacles in comparison to the transient case. Our operators feature potentials W that decay to zero at infinity, and may be critical or subcritical. In this setting we give matching upper and lower bounds for the heat kernel of Schrödinger operators. This is joint work with Laurent Saloff-Coste. Strongly Tail-Optimal Scheduling in the Light-Tailed M/G/1 Ziv Scully (Cornell University) Monday, 3rd of February 2025 at in Malott 406 Probability We study the problem of scheduling jobs in a queueing system, specifically an M/G/1 with light-tailed job sizes, to asymptotically optimize the response time tail. For some time, the best known policy was First-Come First-Served (FCFS), which has an asymptotically exponential tail. FCFS achieves the optimal exponential decay rate, but its leading constant is suboptimal. Designing a policy that minimizes this leading constant is a long-standing open problem.

We solve this open problem with a new scheduling policy called $\gamma$-Boost. Roughly speaking, $\gamma$-Boost operates similarly to FCFS, but it pretends that small jobs arrive earlier than their true arrival times. This reduces the response time of small jobs without unduly delaying large jobs. We prove $\gamma$-Boost's asymptotic tail optimality, and we show via simulation that $\gamma$-Boost has excellent practical performance.

The $\gamma$-Boost policy as described above requires knowledge of job sizes. In preliminary work, we generalize $\gamma$-Boost to work with unknown job sizes, proving an analogous asymptotic optimality result in the unknown-size setting. Our generalization reveals that $\gamma$-Boost is a type of Gittins index policy, but with an unusual feature: it uses a negative discount rate.

Joint work with George Yu (Cornell) and Amit Harlev (Cornell). Wetting and pre-wetting in (2+1)D solid-on-solid interfaces Reza Gheissari (Northwestern University) Monday, 9th of December 2024 at in Malott 406 Probability The $(d+1)D$-solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d \ge 2$, with zero-boundary conditions, at low temperatures the surface is localized about height $0$, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$.  I will describe the mechanism of entropic repulsion, and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves, and for metastability phenomena in the Glauber dynamics. Based on joint work with Eyal Lubetzky and Joseph Chen. Darboux transformation of diffusion processes Alexey Kuznetsov (York University) Monday, 18th of November 2024 at in Malott 406 Probability Darboux transformation is a well-known technique in the study of second-order linear differential operators. Darboux transform of a second-order linear differential operator gives another such operator, which is ​"almost" isospectral to the first one –- the spectrum may be different at most at one point. We will show how to use Darboux transformation to construct new families of diffusion processes whose transition probability densities (and their spectral expansions) are given in closed form. We will also discuss connections to exceptional orthogonal polynomials. This talk is based on joint work with Minjian Yuan. Lower bounds on Lyapunov exponents for linear PDEs driven by random velocity Jaeyun Yi (EPFL (École polytechnique fédérale de Lausanne)) Monday, 11th of November 2024 at in Malott 406 Probability We will discuss the Lyapunov exponent for linear PDEs driven by random velocity and provide its lower bound. We focus on two important models: passive scalar advection and linearized stochastic Naiver-Stokes equation. We will describe two turbulent properties of fluid: mixing and chaos. This talk is based on joint work with Marin Hairer, Sam Punshon-Smith, and Tommaso Rosati. Gaussian fluctuations in the infinite volume limit for the focusing $\Phi^4$ measure in 1D Philippe Sosoe (Cornell University) Monday, 4th of November 2024 at in Malott 406 Probability I will introduce a (non Gaussian) measure on functions on the circle considered by Lebowitz Rose and Speer in the 1980s. This measure was shown by Bourgain in the 1990s to be invariant for the cubic nonlinear Schroedinger equation, a deterministic PDE.

In the limit where the radius of the circle increases to infinity, it was shown by Rider that the measure collapses to a delta function on the zero path. I will explain this result in terms of the optimizers of a certain Sobolev inequality, and describe a new result with Kihoon Seong identifying the fluctuations about this trivial limit as white noise. Clustering of large deviations in heavy tailed moving average process Jiaqi Wang (Cornell University) Monday, 28th of October 2024 at in Malott 406 Probability Large deviation events, like hurricanes or financial crises, can have severe consequences. When these events cluster, occurring close together, their impact intensifies. In our study of the moving average process under heavy-tailed distributions, we analyzed this clustering behavior. We found that the "conspiracy vs. catastrophe" distinction between light and heavy tailed distributions still applies. Using the "single large jump" intuition, we demonstrated that cluster sizes follow an approximately uniform distribution that grows linearly with sample size. Zigzag strategy for random matrices Joscha Henheik (IST Austria) Monday, 21st of October 2024 at in Malott 406 Probability It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.

These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory.

In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices.

Based on joint works with G. Cipolloni, L. Erdős, O. Kolupaiev, and V. Riabov. Why is nature critical? Matthew Junge (Baruch College) Monday, 7th of October 2024 at in Malott 406 Probability How is it that natural systems---such as tectonic plates, snow slopes, and financial markets---exhibit the power-law and fractal behaviors characteristic of a critical state? In 1987, physicists Bak, Tang, and Wiesenfeld developed a widely cited explanation known as Self-Organized Criticality (SOC). I will present Hoffman, Johnson, and my recent article arXiv:2406.01731, in which we give one of the first rigorous mathematical demonstrations of SOC by proving the density conjecture in dimension one for the Activated Random Walk model. Language models for mathematicians Lionel Levine (Cornell University) Monday, 30th of September 2024 at in Malott 406 Probability The output of a language model like ChatGPT is a probability distribution for the next "token" (a short word or fragment of a long word). Mathematically, what is the model actually doing when it generates these probabilities? I’ll narrate how the model translates its input text into a sequence of vectors, which pass through a succession of alternating linear and nonlinear layers. I’ll examine this architecture with a view toward some big-picture questions: Do language models "plan ahead" for future words, phrases, and sentences? And if so, then how does training the model to predict just one token at a time incentivize long-term planning? Based on joint work arXiv:2404.00859 with Wilson Wu and John X. Morris. Novel algebraic perspectives on self-similarity Pierre Patie (Cornell University) Monday, 23rd of September 2024 at in Malott 406 Probability Self-similarity is a pervasive concept in mathematics, appearing in fields as diverse probability and operator theory, and mathematical physics. In this talk, we offer a novel and comprehensive perspective on the rich structure of self-similarity by exploring its connections with group representation theory and operator algebras. We will discuss the surprising roles of the Bessel operator and the Laplacian within this framework. By the end of the talk, we will uncover how these structures enhance our understanding of scaling limits and universality, with a particular focus on the spectrum of large random matrices. Algorithmic threshold for the random perceptron Brice Huang (MIT) Monday, 16th of September 2024 at in Malott 406 Probability We consider the problem of efficiently optimizing random (spherical or Ising) perceptron models with general bounded Lipschitz activation. We focus on a class of algorithms with Lipschitz dependence on the disorder: this includes gradient descent, Langevin dynamics, approximate message passing, and any constant-order method on dimension-free time-scales. Our main result exactly characterizes the optimal value ALG such algorithms can attain in terms of a one-dimensional stochastic control problem. Qualitatively, ALG is the largest value whose level set contains a certain "dense solution cluster." Quantitatively, this characterization yields both improved algorithms and hardness results for a variety of asymptotic regimes, which are sharp up to absolute constant factors. Joint work (in progress) with Mark Sellke and Nike Sun. Infinite geodesics and Busemann functions in inhomogeneous exponential last passage percolation Christopher Janjigian (Purdue University) Monday, 9th of September 2024 at in Malott 406 Probability This talk will discuss some recent progress on understanding the structure of semi-infinite geodesics and their associated Busemann functions in the inhomogeneous exactly solvable exponential last-passage percolation model. In contrast to the homogeneous model, this generalization admits linear segments of the limit shape and an associated richer structure of semi-infinite geodesic behaviors. Depending on certain choices of the inhomogeneity parameters, we show that the model exhibits new behaviors of semi-infinite geodesics, which include wandering semi-infinite geodesics with no asymptotic direction, isolated asymptotic directions of semi-infinite geodesics, and non-trivial intervals of directions with no semi-infinite geodesics. Based on joint work with Elnur Emrah (Bristol) and Timo Seppäläinen (Madison) Sampling from mean-field spin glass Gibbs measures and obstructions Ahmed El Alaoui (Cornell University) Monday, 26th of August 2024 at in Malott 406 Probability I will discuss the question of sampling from the p-spin Gibbs measure. Whether or not this task is efficiently possible is connected to the geometric properties of the measure, which undergoes a sequence of phase transition phenomena as the temperature is changed. I will present the phase digram and discuss what is supposed to happen in each phase. I will then construct an efficient sampling algorithm in a large sub-interval of temperatures where this task is believed to be possible, and rule out a large sub-family of efficient algorithms when it is not. The latter obstruction is obtained by proving strong notions of shattering and chaos for the measure. 2023 — 24 Groups, embeddings, and random walks Ramon van Handel (Princeton University) Monday, 29th of April 2024 at in Malott 406 Probability A finite group with a given set of generators may naturally be viewed as a metric space endowed with the word metric. Understanding its geometry is a basic question of geometric group theory. One may ask, for example, when this metric space can be embedded with low distortion into a normed space (in other words, "can the word metric be approximately described a norm"?) My aim in this talk is to explain a connection between this problem and the properties of random walks on groups. Even the simplest possible example where this connection is fully understood, the discrete cube, settled a long-standing problem of Enflo in the geometry of Banach spaces (joint work with Ivanisvili and Volberg). In other groups, however, this program gives rise to natural questions about the behavior of random walks that do not appear to have been studied in the probability literature. I aim to explain ongoing work with Mira Gordin on the symmetric group, and some challenges we encounter in understanding more general situations. Universality in the Persistence of Random Polynomials Promit Ghosal (Brandeis University) Monday, 22nd of April 2024 at in Malott 406 Probability Consider random polynomials of the form $\sum_{i=0}^n a_i x^i$, where the coefficients $a_i$ are independent with mean zero and finite second moments. We discuss how the probability that such a polynomial has no zeros decays as $n^{-b+o(1)}$, where $b$ is a universal constant independent of the distributions of $a_i$. This resolves a conjecture by Poonen and Stoll. Notable progress on this conjecture was made by Dembo, Poonen, Shao, and Zeitouni, who demonstrated the same result when coefficients are identically and independently distributed (i.i.d.) with all higher moments being finite. Extending their results to the finite moment case remained an open problem until now. Our approach, in contrast to previous works in this area, relies on the combinatorics of multiscale analysis of random polynomials. This work is based on joint collaboration with Sumit Mukherjee. A spectral and algebraic algorithm: the centralizer and the fixed points, scaling and universality classes Pierre Patie (Cornell University) Monday, 15th of April 2024 at in Malott 406 Probability Over the last few decades, the exploration of scaling limits and universality classes has unveiled a spectrum of intriguing results, alongside complex and fascinating challenges. In this talk, we present a comprehensive framework designed to address these challenges in a constructive and solvable manner. It is based on an appropriate combination of group representation theory, group actions, spectral theory and operator algebras. Relying on the Stone-von Neumann Theorem, we identify a canonical setting for this framework, the so-called canonical G-module, and design a constructive mathematical algorithm that reduces this problem to the category 0 (set and functions). This formalism not only highlights the fundamental role played by the choice of the representation of mathematical objects but also offers constructive perspectives and connections into classical mathematical topics such as the spectral theory of self-adjoint operators, Lie point symmetry, von Neumann algebras, and the Stone-von Neumann theorem. We will illustrate this framework by describing the universality classes of the LUE ensemble, emphasizing the canonical role of its limit to the Bessel ensemble in the comprehensive framework. Finally, we shall discuss the role played by the Fourier transform, the Laplacian, and the Brownian motion in this formalism. Jointly invariant measures for the Kardar-Parisi-Zhang equation Evan Sorensen (Columbia University) Monday, 8th of April 2024 at in Malott 406 Probability We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of Janjigian, Rassoul-Agha, and Seppalainen, by showing the existence of a random, countably infinite dense set of directions at which the Busemann process of the KPZ equation is discontinuous. This signals instability and shows the failure of the one force–one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. As the inverse temperature parameter for the KPZ equation $\beta$ goes to $\infty$, the KPZH converges to the stationary horizon (SH), a process which describes the jointly invariant measures for the KPZ fixed point. As $\beta$ approaches $0$, the KPZH converges to a coupling of Brownian motions that differ by linear shifts, which is a jointly invariant measure for the Edwards-Wilkinson fixed point. Joint work with Sean Groathouse, Firas Rassoul-Agha, and Timo Seppäläinen. Matrix displacement convexity and intrinsic dimensionality Yair Shenfeld (Brown University) Monday, 25th of March 2024 at in Malott 406 Probability The space of probability measures endowed with the optimal transport metric has a rich structure with applications in probability, analysis, and geometry. The notion of (displacement) convexity in this space was discovered by McCann, and forms the backbone of this theory. I will introduce a new, and stronger, notion of displacement convexity which operates on the matrix level. The motivation behind this definition is to capture the intrinsic dimensionality of probability measures which could have very different behaviors along different directions in space. I will show that a broad class of flows satisfy matrix displacement convexity: heat flow, optimal transport, entropic interpolation, mean-field games, and semiclassical limits of non-linear Schrödinger equations. This leads to intrinsic dimensional functional inequalities which provide a systematic improvement on numerous classical functional inequalities. Two-point function of KPZ fixed point with Gaussian initial data Sefika Kuzgun (University of Rochester) Monday, 18th of March 2024 at in Malott 406 Probability In this talk, we consider KPZ fixed point starting from a Gaussian process with stationary increments. Using Malliavin integration by parts, we establish the formula for two-point correlation function of the spatial derivative process in terms of the variance of the KPZ fixed point. This talk is based on an ongoing project with Arjun Krishnan. The coupling method for central moment bounds in exponential last-passage percolation Janosch Ortmann (University of Quebec in Montreal) Monday, 4th of March 2024 at in Malott 406 Probability KPZ universality describes a scaling behaviour that differs from the central limit theorem by the size of the fluctuations (cube-root instead of square-root) and the limiting distribution. Instead of the Gaussian, the Tracy-Widom distributions from random matrix theory appear in the limit. It is a long standing conjecture that the KPZ universality class contains a large group of models, including particle systems and polymer models.

A key model in the KPZ universality is last-passage percolation (LPP), i.e. the distribution of maximal weights across directed paths in a random environment. In this talk I will discuss several variants of the LPP model with independent exponential weights. In particular, I will show how probabilistic coupling, originating from work by Cator and Groeneboom on Hammersley’s process and the Poisson LPP, an be used to derive optimal-order upper and lower bounds on the central moment for these two variants of exponential LPP. That is, letting $v$ be the LPP end-point we obtain bounds proportional to $\|v\|^{p/2}$ (CLT scaling) when $v$ is close to the axis and to $\|v\|^{p/3}$ (KPZ regime) otherwise. These bounds are also uniform over vertices taking values in these regions.

The talk is based on joint work with Elnur Emrah and Nicos Georgiou. Yang-Mills theory and random surfaces Minjae Park (University of Chicago) Monday, 19th of February 2024 at in Malott 406 Probability ​I will talk about some recent work on Yang-Mills theory and its connection to the theory of random surfaces. In particular, Wilson loop expectations (important observables in Yang-Mills) can be represented as surface sums in two settings:

​- 2D continuum Euclidean Yang-Mills for classical Lie groups of any matrix size N (based on joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu)

- Any dimensional lattice Yang-Mills for classical Lie groups of any matrix size N and any inverse temperature β (based on joint work with Sky Cao and Scott Sheffield) In addition, our probabilistic approaches provide alternative derivations and interpretations of several recent theorems, including Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov), and surface sums (Magee and Puder). New lower bounds for sphere packings and independence sets via randomness Marcus Michelen (University of Illinois Chicago) Monday, 12th of February 2024 at in Malott 406 Probability We show new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not too large co-degrees. For dimension d, this achieves a sphere packing of density $(1 + o(1)) d \log d / 2^(d+1)$. In general dimension, this provides the first asymptotically growing improvement for sphere packing lower bounds since Roger's bound of $c d/2^d$ in 1947. The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not too large co-degree. Both steps will be discussed, and no knowledge of sphere packings will be assumed or required. This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe. Some rigorous results on the Lévy spin glass model Wei-Kuo Chen (University of Minnesota) Monday, 5th of February 2024 at in Malott 406 Probability The Lévy spin glass model, proposed by Cizeau-Bouchaud, is a mean-field model defined on a fully connected graph, where the spin interactions are formulated through a power-law distribution. This model is well-motivated from the study of the experimental metallic spin glasses. It is also expected to bridge between some mean-field and diluted models. In this talk, we will discuss some recent progress on the Lévy model including its high temperature behavior and the existence and variational expression for the limiting free energy. Based on a joint work with Heejune Kim and Arnab Sen. Stable-like Random Walks on Group of Polynomial Growth Ruoqi Zhang (Cornell University) Monday, 29th of January 2024 at in Malott 406 Probability The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions. The work of Kenyon, Okounkov, Sheffield and others has shown that periodic lattices with non-trivial Newton polygons lead to rich arctic curves with many frozen and gaseous regions. Groves are another model, closely related to spanning trees, that exhibits an arctic circle theorem, due to Petersen and Speyer. We compute arctic curves for groves with non-trivial Newton polygons, and provide a geometric description of asymptotic edge probabilities. Transition State Theory for $\Phi^4_3$ Nikolay Barashkov (Max Planck Institute) Monday, 22nd of January 2024 at in Malott 406 Probability The Wave equation in a double well potential, with random invariant initial data is expected to exhibit metastable behaviour: as the temperature approaches 0, the solution spends exponentially long times near the wells and the behaviour of the expected transition times is to be given by the Eyring Kramers Law. The amount of crossings between the wells is bounded by the amount of crossings of a hyperplane between them. We establish an Eyring-Kramers formula for the hyperplane crossings in 2 and 3 dimensions, building on work of Vanden-Eijnden and Newhall in 1 dimension.

This is joint work with Petri Laarne (University of Helsinki). A Sobolev space theory for SPDEs with space-time non-local operators Junhee Ryu (Brown University) Monday, 4th of December 2023 at in Malott 406 Probability In this talk, I will discuss a Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes as well as the SPDE driven by space-time white noise. For the time non-local operator, we consider the Caputo fractional derivative of order $\alpha\in(0,1)$. For the spatial non-local operator, we deal with the infinitesimal generator of the $d$-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions. Dark matter might be a condensate of nonlinear quantum particles Kay Kirkpatrick (University of Illinois at Urbana-Champaign) Monday, 27th of November 2023 at in Malott 406 Probability Dark matter, which has no color but rather is transparent, makes up probably a quarter of the energy-matter density of our cosmos and has a lot of evidence for its existence, including gravitational lensing and cosmic microwave background fluctuations. One model for dark matter is the hypothetical quantum particle that solves the strong Charge+Parity problem in quantum chromodynamics: the axion. Axions probably clustered in the early universe, condensing into ultra-cold Bose-Einstein Condensates (BECs) that have macroscopic quantum properties. For axion clusters the mathematical description is a coupled pair of PDEs: the non-linear quantum (NLQ) equation and the Poisson equation for interactions through gravity.

We will survey BECs, axions, and the NLQ. In recent work with Partha Dey and Kesav Krishnan, we find a phase transition in the discrete NLQ. In recent work with Anthony Mirasola and Chanda Prescod-Weinstein, we calculate that the condensation of axions into a BEC is driven by gravity and should be possible within the lifetime of our cosmos. Stochastic dynamics and the Polchinski equation Benoit Dagallier (Courant Institute) Monday, 20th of November 2023 at in Malott 406 Probability I will discuss a general framework to obtain large scale information in statistical mechanics and field theory models. The basic idea is to obtain information from the model by constructing a dynamics that samples from it and analysing the long-time behaviour of the dynamics. The Langevin dynamics is a typical example of such a dynamics. In this talk I will introduce another, the Polchinski dynamics, based on renormalisation group ideas. The dynamics is parametrised by a parameter representing scale and has a number of interesting properties that make it well suited to study large-dimensional models. It is perhaps better known under the name stochastic localisation. I will mention a number of recent applications of this dynamics, in particular to prove functional inequalities via a generalisation of Bakry and Emery's convexity-based argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau and the recent review paper arXiv:2307.07619. Active Phase for the Stochastic Sandpile on $\mathbb{Z}$ Douglas Rizzolo (University of Delaware) Monday, 13th of November 2023 at in Malott 406 Probability Sandpile models have a long history in the statistical mechanics literature as paradigms of self-organized criticality. Of particular importance is the Stochastic Sandpile Model. This abelian variant of Manna’s model is widely believed to exhibit universality. Since the Stochastic Standpile Model gained popularity in the mathematics community the prime challenge has been to prove that the critical density for the one-dimensional Stochastic Sandpile Model is less than one. In this talk we will discuss the recent proof of this conjecture. Based on joint work with Christopher Hoffman, Yiping Hu, and Jacob Richey. Convergence problems for singular stochastic dynamics Younes Zine (EPFL (École polytechnique fédérale de Lausanne)) Monday, 6th of November 2023 at in Malott 406 Probability Over the last twenty years there has been significant progress in the well-posedness study of singular stochastic PDEs in both parabolic and dispersive settings. In this talk, I will discuss some convergence problems for singular stochastic nonlinear PDEs. In a seminal work, Da Prato and Debussche (2003) established well-posedness of the stochastic quantization equation, also known as the parabolic $\Phi^{k+1}_2$-model in the two-dimensional case. More recently, Gubinelli, Koch, Oh, and Tolomeo proved the corresponding well-posedness for the canonical stochastic quantization equation, also known as the hyperbolic $\Phi^{k+1}_2$-model in the two-dimensional case. In the first part of this talk, I will describe convergence of the hyperbolic $\Phi^{k+1}_2$-model to the parabolic $\Phi^{k+1}_2$-model. In the dispersive setting, Bourgain (1996) established well-posedness for the dispersive $\Phi^4_2$-model (=deterministic cubic nonlinear Schrödinger equation) on the two-dimensional torus with Gibbsian initial data. In the second part of the talk, I will discuss the convergence of the stochastic complex Ginzburg-Landau equation (= complex-valued version of the parabolic $\Phi^4_2$-model) to the dispersive $\Phi^4_2$-model at statistical equilibrium. Homogeneous approximations of semi-Markov models and applications to life insurance Oscar Peralta (Cornell University) Monday, 30th of October 2023 at in Malott 406 Probability A semi-Markov model is a multi-state jump process in which the jump intensities vary based on the current state, calendar time, and duration since the last jump. In the context of life insurance, these models allow for transitions based on age and time since the last transition between states. In this talk, we demonstrate how they can be accurately approximated, in a pathwise sense, using time-homogeneous Markovian jump processes. These are constructed by observing the original model at an increasingly dense set of Poisson arrivals and placing those observations at a different set of independent Poisson times. The primary advantage of our approximation is the existence of closed-form formulae for its transition probabilities, which are not explicitly available in the exact case. In summary, our method simplifies the study of descriptors of semi-Markov models through approximations with homogeneous Markovian jump processes, offering a valuable tool for pricing life insurance policies under such models. This is joint work with Martin Bladt and Andreea Minca. Veridical Data Science Toward Trustworthy AI Bin Yu (UC Berkeley) Monday, 23rd of October 2023 at in Bill and Melinda Gates, 114 Probability Data Science is central to AI and has driven most of recent advances in biomedicine and beyond. Human judgment calls are ubiquitous at every step of a data science life cycle (DSLC): problem formulation, data cleaning, EDA, modeling, and reporting. Such judgment calls are often responsible for the "dangers" of AI by creating a universe of hidden uncertainties well beyond sample-to-sample uncertainty. To mitigate these dangers, veridical (truthful) data science is introduced based on three principles: Predictability, Computability and Stability (PCS). The PCS framework and documentation unify, streamline, and expand on the ideas and best practices of statistics and machine learning. In every step of a DSLC, PCS emphasizes reality check through predictability, considers computability up front, and takes into account of expanded uncertainty sources including those from data curation/cleaning and algorithm choice to build more trust in data results. PCS will be showcased through collaborative research in finding genetic drivers of a heart disease, stress-testing a clinical decision rule, and identifying microbiome-related metabolite signature for possible early cancer detection. Heat flow, random matrices, and random polynomials Brian Hall (University of Notre Dame) Monday, 16th of October 2023 at in Malott 406 Probability It is an old result of Polya and Benz that applying the backward heat flow to a polynomial with all real zeros gives another polynomial with all real zeros. Much more recently, the limiting behavior of the real zeros as the degree goes to infinity has been worked out, with a surprising connection to random matrix theory. The situation is more complicated if we use the forward heat flow—in which case, the zeros will not remain real—or if we apply the heat flow to a polynomial with complex roots. Nevertheless, there is still a conjectural connection to random matrix theory. Consider, for example, the circular law in random matrix theory: If a random matrix Z has i.i.d. entries, its eigenvalues will be asymptotically uniform over a disk. The heat flow then conjecturally changes the circular law into the elliptical law: Applying the heat flow to the characteristic polynomial of Z should give a new polynomial whose zeros are asymptotically uniform over an ellipse. While the random matrix case remains a conjecture, we have rigorous results for random polynomials with independent coefficients. This is joint work with Ching Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The talk will be self-contained and have lots of pictures and animations. Moment Intermittency in the PAM with Asymptotically Singular Noise Pierre Yves Gaudreau Lamarre (Syracuse University) Monday, 2nd of October 2023 at in Malott 406 Probability The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.

One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.

In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.

This talk is based on a joint work with Promit Ghosal and Yuchen Liao. Threshold for detecting changes in Erdős-Rényi graphs Shirshendu Chatterjee (City College New York) Monday, 25th of September 2023 at in Malott 406 Probability We will discuss the offline change-point detection and localization problem in the context of piece-wise stationary inhomogeneous Erdős-Rényi (ER) random graphs, where the observable is a finite sequence of inhomogeneous ER random graphs. We will discuss the associated challenges, detectability and localizability thresholds, their relationship with the ER random graph sequence parameters, and some of the available algorithms for detecting the change points. Tail estimates for the stationary six vertex model and ASEP Philippe Sosoe (Cornell University) Monday, 11th of September 2023 at in Malott 406 Probability Phase transition issue of focusing Gibbs measures for Schroedinger-wave systems Kihoon Seong (Cornell University) Monday, 28th of August 2023 at in Malott 406 Probability We study the phase transition phenomenon of the singular Gibbs measure associated with the Schroedinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence. The singularity of the Gibbs measure creates an additional difficulty in proving the non-convergence in the strong coupling case. 2022 — 23 Finding the source Jacob Richey (University of British Columbia) Monday, 8th of May 2023 at in Malott 406 Probability Consider a random process on a graph. Given only a snapshot of the set of visited sites at some time, how easy is it to guess the starting point? I will present ideas and problems related to this question in two contexts: disease transmission/rumor spread on a social network, and simple random walk/Brownian motion. Traces, Eigenvalues, and Eigenvectors Simona Diaconu (Stanford University) Monday, 1st of May 2023 at in Malott 406 Probability Random matrix theory is concerned primarily with the eigenvalues and eigenvectors of certain ensembles. The method of moments is a common technique in this area: it goes back to Wigner in 1955, and consists of finding the asymptotic behavior of traces of large powers of matrices. The most common uses are bounds on the operator norm with high probability, a classical result of Soshnikov from 1998 being edge eigenspectrum universality. The purpose of this talk is highlighting the versatility of this technique by presenting three (more) recent applications, two yielding central limit theorems for eigenvalues, and one perhaps even surprising, related to eigenvectors. No background in random matrix theory is assumed. Cluster expansion approach in mean-field disordered models. Partha Dey (University of Illinois Urbana-Champaign) Monday, 24th of April 2023 at in Malott 406 Probability Cluster expansion approach was used in the seminal work of Aizenman-Lebowitz-Ruelle (1987) to understand the distributional limit of the free energy in the high-temperature SK spin glass model with zero external field. The main idea is to convert the probabilistic problem into a counting problem. We use this approach to understand the high-temperature behavior of a few disordered models in detail. Examples include -

a) SK model under weak external field,

b) pure and mixed p-spin glass model with zero and weak external field,

c) heavy-tailed p-spin glass under zero and weak external field,

d) Potts spin glass model,

e) mean field traveling salesman, matching and spanning tree problems.

Based on joint work with Qiang Wu and Greg Terlov. The nonlinear stochastic heat equation in the critical dimension Alex Dunlap (New York University) Monday, 17th of April 2023 at in Malott 406 Probability I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu. Scaling limit of soliton lengths in a multicolor box-ball system Hanbaek Lyu (University of Wisconsin, Madison) Monday, 10th of April 2023 at in Malott 406 Probability The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this talk, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain sharp scaling limit of soliton lengths that turns out to be different from the single color case as established in Levine, Lyu, and Pike ’20. A large part of our analysis is devoted to study the associated carrier process, which is a multi-dimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers, and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of the modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes. Bulk deviation lower bounds for the simple random walk Maximilian Nitzschner (New York University) Monday, 27th of March 2023 at in Malott 406 Probability In this talk we present large deviation lower bounds for the probability of certain bulk-deviation events depending on the occupation-time field of a simple random walk on the Euclidean lattice in dimensions larger or equal to three.

As a particular application, these bounds imply an exact leading order decay rate for the probability of the event that a simple random walk covers a substantial fraction of a macroscopic body, when combined with a corresponding upper bound previously obtained by Sznitman. As a pivotal tool for deriving such optimal lower bounds, we recall the model of tilted walks which was first introduced by Li in order to develop similar large deviation lower bounds for the probability of disconnecting a macroscopic body from an enclosing box by the trace of a simple random walk. We then discuss a refined local coupling with the model of random interlacements which is used to locally approximate the occupation times of the tilted walk.

Based on joint work in progress with A. Chiarini (University of Padova). Homogenization for the random G equation Bill Cooperman (University of Chicago) Monday, 20th of March 2023 at in Malott 406 Probability How can a fish swim from point A to point B if the ocean current flows faster than the fish can swim? If the current is mean-zero and random in space, then the fish has a chance. I will discuss some results on stochastic homogenization of the G equation, a convex but non-coercive Hamilton-Jacobi equation. Along the way, I will present a simplified proof of a quantitative first-passage percolation shape theorem of Kesten. The random matrix Fyodorov-Hiary-Keating conjecture. Elliot Paquette (McGill University) Monday, 13th of March 2023 at in Malott 406 Probability The Fyodorov-Hiary-Keating conjecture has two parts, one in random matrix theory and one about the Riemann zeta function. In the random matrix part, it gives the precise distributional limit for the maximum of a characteristic polynomial of a Haar Unitary matrix. Using the replica method and a physically motivated "freezing" ansatz, they derived one of the most precise log-correlated field predictions to date, and they did it for a process which was not even Gaussian. While existing work shows that Haar Unitary matrices had many log correlated field connections, techniques for showing convergence of the maximum typically rely on either the Gaussianity of the underlying process or precise branching structures built into the problem; the characteristic polynomial has neither. We will describe the problem and the current state of the art, in which we (the speaker and Ofer Zeitouni) show the convergence in law of the maximum of a Circular-beta ensemble random matrix to a convolution of a gumbel and the total mass of a (non-Gaussian) critical multiplicative chaos. Universality and well-posedness for a time-inhomogeneous KPZ equation Kevin Yang (University of California, Berkeley) Monday, 6th of March 2023 at in Malott 406 Probability This talk has two goals. The first is a derivation of a time-inhomogeneous KPZ equation from fluctuations in a Ginzburg-Landau SDE in nonequilibrium. The method is a fluctuation-scale analog of Yau’s method for hydrodynamic limits in nonequilibrium. The second is a (possibly short) discussion about well-posedness of the limit KPZ equation itself, which has a log-nonlinearity that is absent in the time-homogeneous case. A number of questions in the spirit of nonequilibrium statistical mechanics (e.g. hysteresis in the microscopic model) will also be addressed, as well as further extensions of this project. Hamilton-Jacobi scaling limit of Pareto hull peeling Peter Morfe (Max Planck Institute) Monday, 20th of February 2023 at in Malott 406 Probability Nondominated sorting and convex hull peeling are two algorithms for ranking multi-dimensional data sets. In recent years, J. Calder and collaborators have shown that the outputs of both algorithms can be approximated via PDEs when the sample size is large. In joint work with A. Bou-Rabee, we analyze the large-sample limit of a related scheme, called Pareto hull peeling, and show that it, too, is governed by a deterministic PDE. In the talk, I will explain the connections between all three algorithms and give the main ideas of the proofs. Geodesics on random regular graphs and random hyperbolic surfaces Benjamin Dozier (Cornell University) Monday, 13th of February 2023 at in Malott 406 Probability There is a fruitful analogy between random regular graphs and random hyperbolic surfaces (of which there are several different models: random covers, random gluings of ideal triangles, and Weil-Petersson measure). The similarities become apparent when the number of vertices, respectively genus, goes to infinity. I will discuss counting closed geodesics on these objects, in particular the "birthday problem", which asks for the length scale at which closed geodesics become very likely to self-intersect. We solve this problem for the graph case, and discuss progress for hyperbolic surfaces. Based on joint work with Jenya Sapir. Cooperative motion in one dimension Erin Beckman (Utah State University) Monday, 6th of February 2023 at in Malott 406 Probability I will discuss a relationship between recursive distributional equations and convergence results for finite difference schemes of PDEs. We will focus on a family of random processes called cooperative motions, which generalize the simple random walk and the hipster random walk introduced in Addario-Berry et. al. in 2020. Our main theorem is a distributional convergence result for cooperative motion in one dimension. I will highlight some of the novel techniques involved, which rely on connecting these processes to various PDEs, including the parabolic p-Laplace equation and nonlinear Hamilton-Jacobi equations. This talk is based on joint work with Louigi Addario-Berry, Gavin Barill, and Jessica Lin. Characterizing nonamenability through  stochastic domination and finitary factors Gourab Ray (University of Victoria) Monday, 30th of January 2023 at in Zoom Probability Take an Ising model with very low temperature. What is the largest p such that the Ising model dominates Bernoulli percolation with parameter p ? We will show that the answer to this question depends drastically on the geometry of the graph. We also obtain similar results for for two Ising models at very low, but close temperatures.

A process is a finitary factor of iid if it can be written as a measurable and equivariant function of an iid process. As an application of the domination results, we show that the very low temperature Ising model on a nonamenable graph is a finitary factor of iid. This is in stark contrast with the amenable setting, where it is known through a celebrated result of Van Den Berg and Steif that the low temperature Ising model is not a finitary factor of iid. Joint work with Yinon Spinka. Invariant Gibbs measures for the three-dimensional cubic nonlinear wave equation Bjoern Bringmann (Princeton University) Monday, 23rd of January 2023 at in Malott 406 Probability In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic $\Phi^4_3$-model. This result is the hyperbolic counterpart to seminal works on the parabolic $\Phi^4_3$-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. We also connect our theorem with classical and recent developments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a non-technical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we discuss our para-controlled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue. Efficient Mean Estimation with Pure Differential Privacy via a Sum-Of-Squares Exponential Mechanism Gautam Kamath (University of Waterloo) Monday, 5th of December 2022 at in 114 Gates Hall Probability We give the first polynomial-time algorithm to estimate the mean of a $d$-dimensional probability distribution with bounded covariance from ~$O(d)$ independent samples, subject to pure differential privacy. Prior algorithms for this problem either incur exponential running time, require $\Omega(d^{1.5})$ samples, or satisfy only the weaker approximate differential privacy condition. Some interesting connections with robust statistics are uncovered along the way.No familiarity with most of the words in the title (differential privacy, sum of squares, etc.) is assumed. Based on joint work in STOC 2022 with Samuel B. Hopkins and Mahbod Majid. ArXiv preprint available at arXiv:2111.12981. Extremal statistics of quadratic forms of GOE/GUE eigenvectors Benjamin McKenna (Harvard University) Monday, 28th of November 2022 at in Malott 406 Probability We consider quadratic forms evaluated at GOE/GUE eigenvectors, like those studied in the context of quantum unique ergodicity. Under a rank assumption, we show that, in order to compute their extremal statistics, it suffices to replace the eigenvectors with independent Gaussian vectors. By carrying out some representative Gaussian computations, we thus find Gumbel and Weibull limiting distributions for the original problem. Joint work with László Erdős. $C_0$-semigroups on the Weyl Chamber with examples Andrew Chee (Cornell University) Monday, 21st of November 2022 at in Malott 406 Probability Motivated by the ubiquity of determinantal structures in the mathematical physics literature, e.g. random matrices, statistical mechanics models, we propose to revisit and extend the seminal construction of Karlin-McGregor by developing a theory of $C_0$-semigroups (non-necessarily Markovian) on the Weyl Chamber $W(E)$. We take an operator theoretical approach and use the following three ideas:
1) a lifting procedure from E to W(E),
2) some classical and more recently introduced isospectral classifications schemes,
3) another view of semigroups on $W(E)$ than the determinantal one.
We illustrate our approach with some examples such as dynamical versions of the LUE ensemble and continuous and discrete Borodin-Muttalib biorthogonal ensembles and their Boson analogues.

Joint work with Pierre Patie. Ballistic Annihilation Matt Junge (Baruch College) Sunday, 13th of November 2022 at in Malott 406 Probability In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions. Random matrices, random groups, and universality Roger Van Peski (MIT) Monday, 7th of November 2022 at in Malott 406 Probability Random integer matrices have been studied since 1980s work by Cohen-Lenstra and Friedman-Washington as models for random groups in number theory and combinatorics, and present both similarities and differences with real/complex random matrices. In this talk I will give some background on the area and discuss a new universal distribution on collections of abelian groups which arises from random matrix products. Based on a joint work arXiv:2209.14957 with Hoi Nguyen, and some earlier results of arXiv:2011.09356, arXiv:2112.02147. Random dimer coverings of the Aztec diamond with doubly periodic edge weights Tomas Berggren (MIT) Monday, 31st of October 2022 at in Malott 406 Probability This talk will be centered around domino tilings, or dimer coverings, of the Aztec diamond with doubly periodic edge weights. Such model exhibits a rich structure, for instance, in the limit three types of regions may appear, the frozen, rough and smooth regions (also known as solid, liquid and gas regions). We will discuss asymptotic results, both on the macroscopic and microscopic scale.

The asymptotic results rely on an expression of the correlation kernel in terms of a Wiener-Hopf factorization of a matrix valued function, which is defined in terms of the edge weights. If time permits, we will discuss how such Wiener-Hopf factorization sometimes can be obtained in a form that are suitable for asymptotic analysis. Random growth on a random surface Ahmed Bou-Rabee (Cornell University) Monday, 24th of October 2022 at in Malott 406 Probability I will describe the large-scale behavior of a random growth model (Internal DLA) on random planar maps which approximate a random fractal surface embedded in the plane (Liouville quantum gravity, LQG).

No prior knowledge of these objects will be assumed. Joint work with Ewain Gwynne. Edge universality of random normal matrices generalizing to higher dimensions Leslie Molag (Bielefeld University) Monday, 17th of October 2022 at in Malott 406 Probability As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density, and in the off-diagonal case, where the Faddeeva plasma kernel emerges. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in higher dimensional determinantal point processes, defined on $\mathbb{C}^d$. The models under consideration concern higher dimensional generalizations of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. Their average density of points converges to a uniform law on a 2d-dimensional hyperellipsoid. It is on the boundary of this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of my knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. Based on arXiv:2208.12676. Superdiffusivity and localization of continuous polymers Sayan Das (Columbia University) Monday, 3rd of October 2022 at in Malott 406 Probability We study the Continuum Directed Random Polymer (CDRP) which arises as a universal scaling limit of discrete directed polymers. In this talk, I will present some of the recent progress in understanding the geometry of the CDRP. In particular, I will show CDRP exhibits pointwise localization and pathwise tightness under 2/3 scaling, confirming the existing predictions in polymer literature under continuous setting. I will explain how our results also shed light on certain properties of the KPZ equation (free energy of the CDRP) such as ergodicity and limiting Bessel behaviors around the maximum. This is based on two joint works with Weitao Zhu. Large deviations in random matrix theory and number theory Emma Bailey (CUNY) Monday, 26th of September 2022 at in Malott 406 Probability There are many connections between the statistical properties of random unitary matrices and various number theoretic functions including the Riemann zeta function. In particular, central limit theorems exist for both characteristic polynomials of randomly drawn unitary matrices and the Riemann zeta function, giving information on ’typical values’. One might instead be interested in ‘atypical’ behaviour, including questions of extreme values. I will discuss recent work with Louis-Pierre Arguin concerning large deviations of the Riemann zeta function, which coincides with the associated theorem in random matrix theory. Surface Growth and the Six Vertex Model Matthew Nicoletti (MIT) Monday, 19th of September 2022 at in Malott 406 Probability We introduce a family of irreversible growth processes which can be seen as Markov chains on discrete height functions defined on the 2-dimensional ​square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice, and "irreversible" means that the height function has nonzero average drift. The dynamics arise naturally from the Yang--Baxter equation for the six vertex model, namely from a construction called "bijectivisation".​

These dynamics preserve the KPZ phase translation invariant Gibbs measures for the stochastic six vertex model, and we compute the current (the average drift) in each KPZ phase pure state with horizontal slope s. Using this, we analyze the hydrodynamic limit of a non-stationary version of the dynamics acting on quarter plane six vertex configurations. Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices Giorgio Cippolloni (Princeton University) Monday, 12th of September 2022 at in Malott 406 Probability We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalization Hypothesis (ETH), which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in QUE.

The main methods behind the above results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii) energy estimates for SEE. 2021 — 22 Exact Results for the Quantum Benjamin-Ono Equation on the Torus Alexander Moll (University of Massachusetts, Boston) Monday, 9th of May 2022 at in Malott 406 Probability In this talk, we use a fractional Gaussian field on the 1-dimensional real torus to quantize Benjamin-Ono waves and give a dynamical interpretation of a result of Stanley (1989). Precisely, the classical Benjamin-Ono equation on the torus is a non-linear and non-local model of dispersive waves. Starting from the standard Gaussian in the symplectic space of this equation, we construct the quantum Benjamin-Ono equation on the torus without any path integrals. Remarkably, a result of Stanley (1989) for Jack polynomials implies an exact description of the spectrum of the quantum Hamiltonian in this model. We prove that if one considers Bohr-Sommerfeld quantization of the classical Benjamin-Ono multi-phase solutions on the torus found by Satsuma-Ishimori (1979), then the resulting approximation of the quantum spectrum found by Stanley is exact after an explicit renormalization of the coefficient of dispersion predicted by Abanov-Wiegmann (2005). Exponential decay of correlations in finite gauge group lattice gauge theories Sky Cao (Stanford University) Monday, 2nd of May 2022 at in Malott 406 Probability Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari. Structure and stability for sparse exponential random graphs Nick Cook (Duke University) Monday, 25th of April 2022 at in Zoom Probability Exponential random graph models (ERGMs) are parametric families of distributions on graphs that are widely applied in the social sciences. They can be viewed as tilts of the Erdős–Rényi model by a Gibbs weight that depends on counts of small subgraphs such as triangles. While they have some appealing features for doing statistics, sampling from these distributions is challenging in many parameter regimes; in some other regimes, typical samples are indistinguishable from the Erdős–Rényi model, or may even "degenerate" to an almost-empty or almost-full graph. Following an idea of Lubetzky and Zhao for the dense case, we show how to cure the degeneracy phenomenon for sparse ERGMs. We further use recent advances in nonlinear large deviations theory for random hypergraphs to establish the typical macroscopic structure of sparse ERGMs in the ferromagnetic parameter regime. Based on joint works with Amir Dembo and Huy Tuan Pham. A half-space beta distributed random walk in random environment Mark Rychnovsky (University of Southern California) Monday, 18th of April 2022 at in Malott 406 Probability We consider random walks on the nonnegative integers in a space-time dependent random environment. We define transition probabilities are given by independent Beta(µ, µ) distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter η. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter η affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on Z. Lozenge Tilings and the Gaussian Free Field on a Cylinder Marianna Russkikh (Massachusetts Institute of Technology) Monday, 11th of April 2022 at in Malott 406 Probability We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes. Lyapunov Exponents of Random Matrix Products and Brownian Motion on GL(n,C) Andrew Ahn (Cornell University) Monday, 28th of March 2022 at in Malott 406 Probability Consider the discrete-time process formed by the singular values of products of random matrices, where time corresponds to the number of matrix factors. It is known due to Oseledets' theorem that under general assumptions, the Lyapunov exponents converge as the number of matrix factors tend to infinity. In this talk, we consider random matrices with distributional invariance under right multiplication by unitary matrices, which include Ginibre matrices and truncated unitary matrices. The corresponding singular value process is Markovian with additional structure that admits study via integrable probability techniques. In this talk, I will discuss recent results on the Lyapunov exponents in the setting where the number M matrix factors tend to infinity simultaneously with matrix sizes N. When this limit is tuned so that M and N grow on the same order, the limiting Lyapunov exponents can be described in terms of Dyson Brownian motion with a special drift vector, which in turn can be linked to a matrix-valued diffusion on the complex general linear group. We find that this description is universal, under general assumptions on the spectrum of the matrix factors. Scaling limits of the Laguerre unitary ensemble Xuan Wu Monday, 21st of March 2022 at in Malott 406 Probability In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler. Dynamical critical 2d first-passage percolation David Harper (Georgia Institute of Technology) Monday, 14th of March 2022 at in Zoom Probability In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam. The Brownian transport map Dan Mikulincer (Massachusetts Institute of Technology) Monday, 7th of March 2022 at in Malott 406 Probability The existence of a transport map from the standard Gaussian leads to succinct representations for, potentially complicated, measures. Inspired by results from optimal transport, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space. Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz when the target measure satisfies an appropriate convexity assumption. This facilitates the proof of several new functional inequalities. In other settings, where a globally Lipschitz transport map cannot exist, we derive Sobolev estimates which are intimately connected to several famous open problems in convex geometry. Joint work with Yair Shenfeld Large deviations, persistent homology, and Brownian conductors with negative resistance Matthew Kvalheim (University of Pennsylvania) Monday, 21st of February 2022 at in Zoom Probability Many biological and physical systems are well-modeled by Brownian particles subject to gradient dynamics plus noise. Important for many applications is the net steady-state particle current or "flux" enabled by the noise and an additional driving force, but this flux is rarely computable analytically. Motivated by this, I will describe joint work with Yuliy Baryshnikov investigating the steady-state flux of nondegenerate diffusion processes on compact manifolds. Such a flux is associated to each one-dimensional real cohomology class and is equivalent to an asymptotic winding rate of trajectories. When the deterministic part of the dynamics is "gradient-like" in a certain sense, I will describe a graph-theoretic formula for the small-noise asymptotics of the flux (based on an extension of Freidlin-Wentzell large deviations theory). When additionally the deterministic part is locally gradient and close to a generic global gradient, there is a natural flux for which the graph-theoretic formula becomes Morse-theoretic and admits a description in terms of persistent homology. As an application, I will explain the paradoxical "negative resistance" phenomenon in Brownian transport discovered by Cecchi and Magnasco (1996). Discrete Self-similar and some ergodic Markov chains Rohan Sarkar (Cornell University) Monday, 14th of February 2022 at in Malott 406 Probability A continuous time Markov chain $X$ on non-negative integers is called discrete self-similar if for any $p$ in the interval $[0,1]$, it satisfies $\operatorname{Binomial}(X(t,n),p) = X(t,\operatorname{Binomial}(n,p))$ in distribution. In this work, we identify a large class of discrete self-similar Markov chains, which also includes the classical birth-death chains. We show that the semigroups associated to these Markov chains satisfy an intertwining relation with the semigroups of self-similar Markov processes on the positive real line. Because of this fact, these Markov chains converge to the self-similar Markov processes on the positive real line after scaling appropriately. By a linear perturbation of the generator of the discrete self-similar Markov chains, we obtain a class of ergodic Markov chains that are non-reversible. In this talk, I will explain how we obtain the spectral properties, entropy decay and hypercontractivity phenomenon of these Markov chains using the concept of intertwining and interweaving relationship. This is a joint work with Laurent Miclo (CNRS and University of Toulouse) and Pierre Patie (Cornell University). The stochastic heat equation with multiplicative Lévy noise: Existence, intermittency, and multifractality Carsten Chong (Columbia University) Monday, 7th of February 2022 at in Malott 406 Probability Gibbsian line ensembles and beta-corners processes Evgeni Dimitrov (Columbia University) Monday, 6th of December 2021 at in Malott 406 Probability Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.

Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.

Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel. Covering systems of congruences Robert Hough (Stony Brook University) Monday, 29th of November 2021 at in Malott 406 Probability A distinct covering system of congruences is a list of congruences $$ a_i \bmod m_i, \qquad i = 1, 2, \dotsc , k $$ whose union is the integers. Erdős asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem. Random planar geometry and the directed landscape Duncan Dauvergne (Princeton University) Monday, 22nd of November 2021 at in Zoom Probability Models of random interface growth and random planar metrics in the KPZ universality class are expected to converge to a universal limit under rescaling. This universal limit is the directed landscape, a scale-invariant random directed metric on the plane, recently constructed by myself, J. Ortmann, and B. Virag. In this talk I will introduce this object and describe its construction. Chemical distance for 2d critical percolation and random cluster model Lily Reeves (Cornell University) Monday, 8th of November 2021 at in Malott 406 Probability In 2-dimensional critical percolation, with positive probability, there is a path that connects the left and right side of a square box. The chemical distance is the expected length of the shortest such path conditional on its existence. In this talk, I will introduce the best known estimates for chemical distance. I will then discuss analogous estimates for the radial chemical distance (the expected length of the shortest path from the origin to distance n), as well as on-going work to extend these estimates to the random cluster model. A portion of this talk is based on joint work with Philippe Sosoe. Card guessing and a variant of the birthday problem Jimmy He (Massachusetts Institute of Technology) Monday, 1st of November 2021 at in Malott 406 Probability Consider a uniformly random deck consisting of cards labelled by numbers from 1 through n, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the game continues. What is the expected number of correct guesses under the best and worst strategies? I will discuss recent work establishing sharp asymptotics for both strategies. For the worst case, this answers a recent question of Diaconis, Graham, He and Spiro. As part of the proof, we study a variant of the birthday problem for sampling without replacement using Stein’s method. This work is joint with Andrea Ottolini. Edge Statistics for Lozenge Tilings of Polygons Jiaoyang Huang (New York University) Monday, 25th of October 2021 at in Malott 406 Probability A central feature of random lozenge tilings is that they exhibit boundary induced phase transitions. Depending on the shape of the domain, they can admit frozen regions and liquid regions. The curve separating these two phases is called the arctic boundary. For random lozenge tilings of polygonal domains, it is predicted that the arctic boundary after proper rescaling converges to the Airy process, a universal scaling limit that is believed to govern various phenomena related to the Kardar–Parisi–Zhang universality class. In this talk, I will give an overview of this edge universality phenomenon, and explain a proof for random lozenge tilings of simply connected polygons forbidding specific (presumably non-generic) behaviors for singular points of the limiting arctic boundary. This is a joint work with Amol Aggarwal. KPZ universality of random growing interfaces Konstantin Mateski (Columbia University) Monday, 18th of October 2021 at in Malott 406 Probability The KPZ universality class includes random growing interfaces, which, after rescaling, are conjectured to converge to the KPZ fixed point. The latter is a Markov process, which has been characterized through the exact solution of TASEP, a particular model in the class. The KPZ equation plays a special role and is conjectured to be the only model connecting the Edwards-Wilkinson (Gaussian) and the KPZ fixed points. In the talk, I will introduce the KPZ fixed point and review recent progress on the KPZ universality. The limit shape of the Leaky Abelian Sandpile Model Sevak Mkrtchyan (University of Rochester) Monday, 4th of October 2021 at in Malott 406 Probability The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1 - 1/d$ of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case $d = 1$. Exact sampling and mixing time of Activated Random Walk Feng Liang (Cornell University) Monday, 27th of September 2021 at in Malott 406 Probability Activated Random Walk is an interacting particle system on the $d$-dimensional lattice $\mathbb{Z}^d$. On a finite subset $V$ of $\mathbb{Z}^d$ it defines a Markov chain on the set of all configurations $V \to \{0,1\}$. We prove that when $V$ is a Euclidean ball intersected with $\mathbb{Z}^d$, the mixing time is at most $(1+o(1))$ times the volume of the ball. The proof uses an exact sampling algorithm for the stationary distribution, a coupling with internal DLA, and an upper bound on the time when internal DLA fills the entire ball. We conjecture cutoff at time $\zeta$ times the volume of the ball, where $\zeta < 1$ is the limiting density of the stationary state. Joint work with Lionel Levine. Singularities in the spectrum of random block matrices David Renfrew (Binghamton University) Monday, 20th of September 2021 at in Malott 406 Probability We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up. Sorting probability for Young diagrams Swee Hong Chan (University of California, Los Angeles) Monday, 13th of September 2021 at in Malott 406 Probability Can you always find two elements $x$, $y$ of a partially ordered set, such that, the probability that $x$ is ordered before $y$ when the poset is ordered randomly, is between $1/3$ and $2/3$? This is the celebrated $1/3$-$2/3$ Conjecture, which has been called "one of the most intriguing problems in the combinatorial theory of posets". We will explore this conjecture for posets that arise from (skew-shaped) Young diagrams, where total orderings of these posets correspond to standard Young tableaux. We will show that that these probabilities are arbitrarily close to $1/2$, by using random walk estimates and the state-of-the-art hook-length formulas of Naruse. This is a joint work with Igor Pak and Greta Panova. This talk is aimed at a general audience. 2020 — 21 Scaling limits of three-dimensional uniform spanning trees and loop-erased random walks Sarai Hernandez-Torres (Technion – Israel Institute of Technology) Monday, 10th of May 2021 at in Zoom Probability The uniform spanning tree (UST) on $\mathbb{Z}^3$ is the infinite-volume limit of uniformly chosen spanning trees of large finite subgraphs of $\mathbb{Z}^3$. The main theorem in this talk is the existence of subsequential scaling limits of the UST on $\mathbb{Z}^3$. We get convergence over dyadic subsequences. An essential tool is Wilson’s algorithm, which samples uniform spanning trees by using loop-erased random walks (LERW). This strategy imposes a restriction: results for the scaling limit of 3D LERW constrain the corresponding results for the 3D UST. We will comment on work in progress for the LERW that leads to the full convergence to the scaling limit of the UST.

This talk is based on joint work with Omer Angel, David Croydon, and Daisuke Shiraishi; and work in progress with Xinyi Li and Daisuke Shiraishi. Long Memory and Conservative Flows on Multiple Stochastic Integrals Shuyang Bai (University of Georgia) Monday, 3rd of May 2021 at in Zoom Probability A stationary sequence is said to have long memory, if the dependence decays slowly so that some non-standard scaling limit arises. Recently, there has been considerable interest in a class of long-memory models involving conservative flows in infinite ergodic theory. In this talk, we shall discuss a class of stationary sequences constructed using conservative flows as well as multiple stochastic integrals. In particular, we will introduce some recent progress on the limit theorems for sums of such sequences. The talk is based on joint works with Takashi Owada, Murad Taqqu, and Yizao Wang. The Airy difference profile and Brownian local time Milind Hegde (University of California- Berkeley) Monday, 26th of April 2021 at in Zoom Probability There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of a canonical limiting object, the parabolic Airy sheet, by Dauvergne-Ortmann-Virág [DOV]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes---a universal limiting geodesic weight profile in planar last passage percolation models---and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important information about the coupling. This difference profile was first studied by Basu, Ganguly, and Hammond, who showed that it is non-decreasing and almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension $1/2$. This also being the Hausdorff dimension of the zero set of Brownian motion, we are led to ask: is there a connection between the two objects? This talk will elucidate such a connection on both local and global scales, making use of the representation of the parabolic Airy sheet via a continuous counterpart of the RSK correspondence as introduced in [DOV]. Tails of the empirical distribution on a geodesic in first-passage percolation Christopher Janjigian (Purdue University) Monday, 19th of April 2021 at in Zoom Probability First-passage percolation defines a random pseudo-metric on $\mathbb{Z}^d$ by attaching to each nearest-neighbor edge of the lattice a non-negative weight. Geodesics are paths which realize the distance between sites. This project considers the question of what the environment looks like on a geodesic through the lens of the empirical distribution of the weights on that geodesic. We obtain upper and lower tail bounds for the upper and lower tails which quantify and limit the intuitive statement that the typical weight on a geodesic should be small compared to the marginal distribution of an edge weight.

Based on joint work-in-progress with Michael Damron, Wai-Kit Lam, and Xiao Shen which was started at the AMS MRC on Spatial Stochastic Models in 2019. Contact process on trees Zoe Huang (Duke University) Monday, 12th of April 2021 at in Zoom Probability The contact process describes an epidemic model where each infected individual recovers at rate 1 and infects its healthy neighbors at rate $\lambda$. We investigate the contact process on inhomogeneous trees including periodic trees and Galton-Watson trees. For the contact process on periodic trees we prove sharp asymptotics for the critical values and the existence of an intermediate phase. For the contact process on Galton-Watson trees, we show that when the offspring distribution (i) is subexponential the critical value for local survival $\lambda_2=0$ and (ii) when it is Geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. Recently it is proved by Bhamidi, Nam, Nguyen and Sly (2019) that when the offspring distribution of the Galton-Watson tree has exponential tail, the first critical value $\lambda_1$ of the contact process is strictly positive. We prove that if the contact process survives then the number of infected sites grows exponentially fast. As a consequence we show that the contact process dies out at the critical value $\lambda_1$ and does not survive strongly at $\lambda_2$. This talk is based on joint work with Rick Durrett. Plaquette Percolation on the Torus Benjamin Schweinhart (University at Albany) Monday, 5th of April 2021 at in Zoom Probability Paul Duncan, Matthew Kahle, and Benjamin Schweinhart

We study plaquette percolation on a $d$-dimensional torus $\mathbb{T}^d_N$, defined by identifying opposite faces of the cube $[0,N]^d,$ and phase transitions marked by the appearance of giant (possibly singular) submanifolds that span the torus. The model we study starts with the complete $(i-1)$-dimensional skeleton of the cubical complex $\mathbb{T}^d_N$ and adds $i$-dimensional cubical plaquettes independently with probability $p$. Our main result is that if $d = 2i$ is even and $\phi_*:H_{i}\left(P;\mathbb{Q}\right)\rightarrow H_{i}\left(\mathbb{T}^d;\mathbb{Q}\right)$ is the map on homology induced by the inclusion $\phi: P \hookrightarrow \mathbb{T}^d$, then $$\mathbb{P}_p\left(\text{$\phi_*$ is nontrivial}\right) \rightarrow \begin{cases} 0 & \text{ if } p<\frac{1}{2} \\ 1 & \text{ if } p>\frac{1}{2} \\ \end{cases}$$ as $N \rightarrow \infty$. For the case $i=2$ and $d=4$ this implies that (possibly singular) surfaces spanning the $4$-torus appear abruptly at $p = 1/2$. We also show that $1$-dimensional and $(d-1)$-dimensional plaquette percolation on the torus exhibit similar sharp thresholds at $\hat{p}_c$ and $1-\hat{p}_c$ respectively, where $\hat{p}_c$ is the critical threshold for bond percolation on $\mathbb{Z}^d$, as well as analogous results for a site percolation model on the torus. Cooperative Motion Random Walks Jessica Lin (McGill University) Monday, 29th of March 2021 at in Zoom Probability We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which describe a type of random walk with dependent delay. The model generalizes the "totally asymmetric lazy hipster random walk" studied by Addario-Berry et al [PTRF, '20]. We introduce a novel approach which relies on convergence results for finite difference schemes of Hamilton-Jacobi equations. This talk is based on joint work with Louigi Addario-Berry and Erin Beckman. Persistence exponent for Gaussian Stationary Processes Sumit Mukherjee (Columbia University) Monday, 22nd of March 2021 at in Zoom Probability In this talk we will study asymptotics of the persistence probability that a Gaussian Stationary Process(GSP) stays positive for a long time interval. If the GSP has non negative correlation function, it follows immediately from Slepian's lemma that there is a persistence exponent. However, the same question has remained entirely open for general GSPs, for almost half a century. In this talk we will discuss this question from a spectral perspective, by studying the underlying spectral measure whose Fourier transform is the correlation function. Developing tools along the way, we will establish the existence of the persistence exponent under mild natural conditions on the spectral measure. We will also demonstrate continuity of the persistence exponent under convergence of spectral measures, in an appropriate metric.

This talk is based on joint work with Naomi D. Feldheim and Ohad N. Feldheim. General solutions and invariant measures for systems of KdV- and Toda-type locally-defined dynamics Makiko Sasada (University of Tokyo/ NYU) Monday, 7th of December 2020 at in Zoom Probability In this talk, I will introduce infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. These systems are understood as "deterministic vertex model", which are discretely indexed in space and time, and their deterministic dynamics is defined locally via lattice equations. They have another formulation via the generalized Pitman’s transform, which is a new crucial observation. We show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Also, a detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. This talk is based on a joint work with David Croydon and Satoshi Tsujimoto. Algorithmic Pure States for the Negative Spherical Perceptron Mark Sellke (Stanford University) Monday, 23rd of November 2020 at in Zoom Probability We consider the spherical perceptron with Gaussian disorder. This is the set $S$ of points $\mathbf{\sigma} \in R^N$ on the sphere of radius $\sqrt{N}$ satisfying $\langle \mathbf{g}_a , \mathbf{\sigma} \rangle \ge k\sqrt{N}\,$ for all $1 \le a \le M$, where $(\mathbf{g}_a)_{a=1}^M$ are independent standard gaussian vectors and $k \in R$ is fixed. In other words it is an intersection of random half-spaces which do not contain the origin when $k>0$ and which do contain the origin when $k\leq 0$. Various characteristics of $S$, such as its surface measure and the largest $M$ for which it is non-empty, were computed heuristically in statistical physics in the asymptotic regime $N \to \infty$, $M/N \to \alpha$. Rigorous results for these questions have been achieved when $k\geq 0$. In this regime, the problem has an "as simple as possible" replica symmetric structure. Moreover from an algorithmic point of view, the problem of computing a point in $S$ reduces to a convex optimization problem. By contrast, in the case $k<0$, $S$ is conjectured to exhibit a hierarchical tree-like geometry known as full replica-symmetry breaking (FRSB) close to the satisfiability threshold $\alpha_{{\rm\tiny SAT}}(k)$, with characteristics captured by a Parisi variational principle akin to the one appearing in the Sherrington-Kirkpatrick model.

In this work we exhibit an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured FRSB structure for $k<0$ and outputs a vector $\hat{\mathbf{\sigma}}$ satisfying $\langle \mathbf{g}_a , \hat{\mathbf{\sigma}}\rangle \ge k \sqrt{N}$ for all $1\le a \le M$ and lying on a sphere of non-trivial radius $\sqrt{\bar{q} N}$, where $\bar{q} \in (0,1)$ is the right-end of the support of the associated Parisi measure. We expect $\hat{\mathbf{\sigma}}$ to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that $\bar{q} \to 1$ as $\alpha \to \alpha_{{\rm\tiny SAT}}(k)$, so that $\big\langle \mathbf{g}_a,\frac{\hat{\mathbf{\sigma}}}{|\hat{\mathbf{\sigma}}|}\big\rangle \geq (k-o(1))\sqrt{N}$ near criticality. This is joint work with Ahmed El Alaoui. Random Matrix Statistics though Pseudo-Randomness Arka Adhikari (Harvard University) Monday, 2nd of November 2020 at in Zoom Probability We introduce the $N \times N$ random matrices $$X_{j,k}=\exp\left(2\pi i \sum_{q=1}^d\ \omega_{j,q} k^q\right) \quad \text{with } \{\omega_{j,q}\}_{\substack{1\leq j\leq N\\ 1\leq q\leq d}} \text{ i.i.d. random variables},$$ where $d$ is a fixed integer. We prove that the distribution of their singular values converges to the local Marchenko-Pastur law at scales $N^{-\theta_d}$ for an explicit, small $\theta_d>0$, as long as $d\geq 18$. To our knowledge, this is the first instance of a random matrix ensemble that is explicitly defined in terms of only $O(N)$ random variables exhibiting a universal local spectral law. Our main technical contribution is to derive concentration bounds for the Stieltjes transform that simultaneously take into account stochastic and oscillatory cancellations. Important ingredients in our proof are strong estimates on the number of solutions to Diophantine equations (in the form of Vinogradov's main conjecture recently proved by Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward identity with an algebraic uniqueness condition for Diophantine equations derived from the Newton-Girard identities.

This is joint work with Marius Lemm. 2019 — 20 Title TBA [CANCELLED] Andrew Ahn (MIT) Monday, 4th of May 2020 at in Malott 406 Probability Topic TBA [CANCELLED] James Michael Leahy (Imperial College London) Monday, 27th of April 2020 at in Malott 406 Probability Title TBA [CANCELLED] Jessica Lin (McGill University) Monday, 20th of April 2020 at in Malott 406 Probability Topic TBA [CANCELLED] Mark Sellke (Stanford University) Monday, 13th of April 2020 at in Malott 406 Probability Title TBA Adhikari Arka (Harvard University) Monday, 6th of April 2020 at in Malott 406 Probability Title TBA Zaoli Chen (Cornell University) Monday, 23rd of March 2020 at in Malott 406 Probability Title TBA Oanh Nyugen (Princeton University) Monday, 16th of March 2020 at in Malott 406 Probability Phase transitions in generalized linear models Leo Miolane (NYU) Monday, 9th of March 2020 at in Malott 406 Probability This is joint work with Jean Barbier, Florent Krzakala, Nicolas Macris and Lenka Zdeborova.

We consider generalized linear models (GLMs) where an unknown n-dimensional signal vector is observed through the application of a random matrix and a non-linear (possibly probabilistic) componentwise output function.

We study the models in the high-dimensional limit, where the observation consists of m points, and m/n→α>0 as n→∞. This situation is ubiquitous in applications ranging from supervised machine learning to signal processing.

We will analyze the model-case when the observation matrix has i.i.d.\ elements and the components of the ground-truth signal are taken independently from some known distribution.

We will compute the limit of the mutual information between the signal and the observations in the large system limit. This quantity is particularly interesting because it is related to the free energy (i.e. the logarithm of the partition function) of the posterior distribution of the signal given the observations. Therefore, the study of the asymptotic mutual information allows to deduce the limit of important quantities such as the minimum mean squared error for the estimation of the signal.

We will observe some phase transition phenomena. Depending on the noise level, the distribution of the signal and the non-linear function of the GLM we may encounter various scenarios where it may be impossible / hard (only with exponential-time algorithms) / easy (with polynomial-time algorithms) to recover the signal. Asymptotic behaviour of high Gaussian minima Arijit Chakrabarty (Indian Statistical Institute) Monday, 2nd of March 2020 at in Malott 406 Probability I shall talk about what happens when an entire sample path of a Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level. The method of answering the questions are different for smooth and non-smooth processes. The former case is analysed in much more details than the latter. On Stationary DLA Jiayan Ye (Texas A&M University) Monday, 17th of February 2020 at in Malott 406 Probability Diffusion limited aggregation (DLA) in $\mathbb{Z}^2$ is a stochastic process first defined by Witten and Sander in order to study aggregation systems governed by diffusive laws. In this talk, I will construct an infinite stationary DLA on the upper half planar lattice. The stationary DLA is ergodic with respect to integer left-right translations. This is a joint work with Eviatar Procaccia and Yuan Zhang. Recent results on the phase transition for activated random walk Jacob Richey (University of Washington) Monday, 10th of February 2020 at in Malott 406 Probability In this talk I will discuss activated random walk, an interacting particle system that exhibits a phase transition on infinite domains and self-organized criticality on finite domains. In the infinite version, the system is initialized with density $\mu$ of particles which perform independent simple random walk, fall asleep at rate $\lambda$, and are woken up if another particle moves to the same site. There are two possible limiting behaviors: local fixation, where each site is visited finitely many times a.s., or non-fixation, where each site is visited infinitely many times a.s. Current research is focused on determining where the transition between these two phenomena occurs in terms of $\mu$ and $\lambda$, and many questions still remain – even on $\mathbb{Z}$. I will present some recent results and discuss the novel tools involved. Arctic curves for groves Terrence George (Brown University) Monday, 3rd of February 2020 at in Malott 406 Probability The limit shape phenomenon is a "law of large numbers" for random surfaces: the random surface looks macroscopically like the "average surface". The first result of this kind was the celebrated arctic circle theorem for domino tilings of the aztec diamond. The limit shape has macroscopic regions with different qualitative behavior, and the arctic curve is the boundary separating these regions. The work of Kenyon, Okounkov, Sheffield and others has shown that periodic lattices with non-trivial Newton polygons lead to rich arctic curves with many frozen and gaseous regions. Groves are another model, closely related to spanning trees, that exhibits an arctic circle theorem, due to Petersen and Speyer. We compute arctic curves for groves with non-trivial Newton polygons, and provide a geometric description of asymptotic edge probabilities. Mean-field disordered systems and Hamilton-Jacobi equations JC Mourrat (NYU) Monday, 27th of January 2020 at in Malott 406 Probability Spin glasses are the most basic examples of disordered systems of statistical mechanics with mean-field interactions. The infinite-volume limit of their free energies are described by the celebrated Parisi formula. I will describe a connection between this result and certain Hamilton-Jacobi equations. The talk will then mostly focus on a simpler setting arising from the problem of inferring a large rank-one matrix, in which the corresponding Hamilton-Jacobi equation is posed in a finite-dimensional space. On the model of non-overlapping hard-disks in discrete $2D$ Izabella Stuhl (Penn State University) Monday, 9th of December 2019 at in Malott 406 Probability An outstanding open problem in statistical mechanics is to determine if the system of identical non-overlapping hard disks in the plane admits a unique Gibbs measure at high-densities. In physical terms, the question is about the existence of a phase transition in this system. A discretization of this problem is a system of hard disks of a given Euclidean diameter $D$ on a unit triangular lattice $A_2$ and a unit square lattice $\mathbb{Z}^2$. A natural view is that, as $D$ tends to infinity, the discrete model (under an appropriate scaling) approaches the continuous one. However, our results show that the situation is more subtle. We give a complete solution, for a general value of $D$, of both discrete versions of model, on $A_2$ and $\mathbb{Z}^2$; in the latter case – in absence of sliding. The main tool is the Pirogov–Sinai theory: we determine the structure of high-density Gibbs measures in the model, relating it to periodic ground states. The answers depend on arithmetic properties of the value $D$ and are given in terms of Eisenstein primes for $A_2$ and solutions to norm equations in the cyclotomic integer ring $\mathbb{Z}[\zeta]$ for $\mathbb{Z}^2$, where $\zeta$ is a primitive 12th root of unity. The number of high-density Gibbs measures in both cases grows indefinitely with $D$ but non-monotonically. This is a joint work with A. Mazel and Y. Suhov. Phase transitions in generalized linear models Leo Miolane (NYU) Monday, 2nd of December 2019 at in Malott 406 Probability This is joint work with Jean Barbier, Florent Krzakala, Nicolas Macris and Lenka Zdeborova.

We consider generalized linear models (GLMs) where an unknown $n$-dimensional signal vector is observed through the application of a random matrix and a non-linear (possibly probabilistic) componentwise output function.

We study the models in the high-dimensional limit, where the observation consists of $m$ points, and $m/n \to \alpha > 0$ as $n \to \infty$. This situation is ubiquitous in applications ranging from supervised machine learning to signal processing.

We will analyze the model-case when the observation matrix has i.i.d. elements and the components of the ground-truth signal are taken independently from some known distribution.

We will compute the limit of the mutual information between the signal and the observations in the large system limit. This quantity is particularly interesting because it is related to the free energy (i.e. the logarithm of the partition function) of the posterior distribution of the signal given the observations. Therefore, the study of the asymptotic mutual information allows to deduce the limit of important quantities such as the minimum mean squared error for the estimation of the signal.

We will observe some phase transition phenomena. Depending on the noise level, the distribution of the signal and the non-linear function of the GLM we may encounter various scenarios where it may be impossible / hard (only with exponential-time algorithms) / easy (with polynomial-time algorithms) to recover the signal. Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential Atilla Yilmaz (Temple University) Monday, 25th of November 2019 at in Malott 406 Probability I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies. Moments of the Riemann Zeta Function on Short Intervals Louis-Pierre Arguin (CUNY Baruch) Monday, 18th of November 2019 at in Malott 406 Probability The moments of the Riemann zeta function on the interval $[T,2T]$ on the critical line play a fundamental role in the distribution of the prime numbers. In this talk, we look at the moments of the Riemann zeta function on typical short intervals, but with length diverging with $T$. We will show that the moments exhibit a freezing phase transition up to a certain interval length akin to the transition seen in $\log$-correlated processes. As a consequence we prove the leading order of the maximum of the zeta function on such short intervals. The results generalize a conjecture of Fyodorov & Keating and the related results of Arguin et al. and Najnudel on intervals of length one.

Joint work with F. Ouimet and M. Radziwill. Gaussian fluctuations in two-dimensional surface growth models Yu-Ting Chen (University of Victoria) Monday, 4th of November 2019 at in Malott 406 Probability In this talk, I will discuss stochastic surface growth dynamics within the

anisotropic class of the Kardar–Parisi–Zhang equation. Fluctuations of the dynamics are conjectured to be Gaussian in the limit of large time as if an unusual nonlinear term in the equation does not exist. The proofs obtained recently consider iterated scaling limits of some particle systems. I will discuss some aspects of these results and explain the particular role of the classical central limit theorem for the problem. An algebraic inverse theorem for the quadratic Littlewood-Offord problem Lisa Sauermann (Stanford University) Monday, 28th of October 2019 at in Malott 406 Probability Consider a quadratic polynomial $f$ in $n$ variables. Furthermore, consider independent Rademacher random variables $\xi_1, \dotsc, \xi_n$ (this means that $Pr(\xi_i=1) = Pr(\xi_i = -1) = 1/2$ for each $i$). How large can the concentration probabilities $Pr(f(\xi_1,\dotsc,\xi_n)=x$) on any single value $x$ be? This is a variant of the classical Littlewood-Offord problem, which asks the same question for a linear polynomial $f$. As in the linear case, it is known that the point probabilities $Pr(f(\xi_1,\dotsc,\xi_n)=x$) can be as large as about $1/\sqrt{n}$. However, for a generic polynomial $f$ with, say, bounded integer coefficients, one would expect the maximum point probabilities to be roughly $1/n$.

In this talk, we discuss an inverse theorem for this problem. More specifically, we show that if $f$ has some point probability larger than $n^{-1+o(1)}$, then $f$ must be close to a quadratic form of bounded rank. For the linear Littlewood-Offord problem there has been an intensive line of research on inverse theorems, with important applications to random matrix theory. Similarly, results on the quadratic Littlewood-Offord problem have had applications to the theory of random symmetric matrices.

Joint work with Matthew Kwan. Eigenvalues for random matrices in the general linear group Brian Hall (University of Notre Dame) Monday, 21st of October 2019 at in Malott 406 Probability One of the most basic examples in random matrix theory is the Ginibre ensemble, in which all the entries of an $N \times N$ matrix are chosen independently with a normal distribution of mean zero and variance $1/N$. When $N$ is large, the eigenvalues are, with high probability, almost uniformly distributed over the unit disk in the complex plane.

One can also think of the Ginibre ensemble as the distribution at time 1 of a Brownian motion in the space of all $N \times N$ matrices. I will then discuss a "multiplicative" analog of this model, consisting of Brownian motion in the group of invertible matrices. In this case, the eigenvalues cluster for large $N$ into a certain region $\Sigma_t$ introduced by Biane. I will describe these regions and the distribution of eigenvalues in them. The talk will be self-contained and have lots of pictures. This is joint work with Bruce Driver and Todd Kemp. Iterative Collaborative Filtering for Sparse Noisy Tensor Estimation Christina Yu (Cornell University) Monday, 7th of October 2019 at in Malott 406 Probability We present a generalization of the collaborative filtering algorithm for the task of tensor estimation, i.e. estimating a low-rank $3$-order $n$-by-$n$-by-$n$ tensor from noisy observations of randomly chosen entries in the sparse regime. Not only does the algorithm have desirable computational properties, it also provably achieves sample complexity that (nearly) matches the conjectured lower bound on the sample complexity. Furthermore, our analysis results in high probability bounds on the infinity norm of the error, as opposed to the weaker MSE bounds achieved by previous approaches. Our proposed algorithm uses the matrix obtained from the "flattened" tensor to compute similarity with respect to a corresponding observation graph. The algorithm recovers the tensor with max entrywise error decaying to $0$ with high probability as long as the entries are sampled uniformly at a density of $\Omega(n^{-3/2 + \varepsilon})$ for any arbitrarily small $\varepsilon > 0$. This sample complexity threshold (nearly) matches a conjectured lower bound as well as the "connectivity threshold" of the corresponding observation graph used in our algorithm, providing a different angle to explain the conjectured lower bound. Matching Random Points Alexander Holroyd (University of Washington) Monday, 30th of September 2019 at in Malott 406 Probability How effectively can two random sets of points (say red and blue) be matched to each other? What can be said about optimal matchings in which the lengths of the edges are minimized? The answer depends in part on precisely what notion of minimality we use. For instance, we may imagine the points as agents who try to greedily optimize their own self-interest, or we may instead aim for global notions of fairness. How do the answers change if we insist that the matching can be determined locally, without global coordination? I will explain how such questions can be turned into formal mathematical ones. I will talk about recent progress and the many open questions, as well as variants such as fair allocation, and multi-color and multi-edge matching.

See http://aeholroyd.org/match/ for pictures. Scaling limits of sandpiles Ahmed Bou-Rabee (University of Chicago) Monday, 23rd of September 2019 at in Malott 406 Probability The Abelian sandpile is a simple combinatorial model from statistical physics which produces striking fractal-like patterns. Why do these patterns appear? What aspects of the patterns persist under the introduction of randomness?

I will introduce the model and then hint at how tools from elliptic partial differential equations and ergodic theory can be used to (partially) answer these questions. Choquet-Deny groups and the infinite conjugacy class property Omer Tamuz (California Institute of Technology) Monday, 16th of September 2019 at in Malott 406 Probability Joint work with Joshua Frisch, Yair Hartman, and Pooya Vahidi Ferdowsi.

The Poisson boundary of a random walk on a group captures the uncertainty in the walk's asymptotic behavior. It has long been known that all commutative groups are Choquet-Deny groups: namely, they have trivial Poisson boundaries for every random walk. More generally, it has been shown that all virtually nilpotent groups are Choquet-Deny. We know that in the class of finitely generated groups, only virtually nilpotent groups are Choquet-Deny. This proves a conjecture of Kaimanovich and Vershik (1983), who suggested that groups of exponential growth are not Choquet-Deny. Our proof does not use the superpolynomial growth property of non-virtually nilpotent groups, but rather that they have quotients with the infinite conjugacy class property (ICC). Indeed, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients. Stable Levy motion with values in the Skorohod space: construction and approximation Raluca Balan (University of Ottawa) Monday, 9th of September 2019 at in Malott 406 Probability Regularly varying random variables play an important role in probability theory, being used as models for heavy-tailed observations (observations which may assume extreme values with high probability). With the rapid advancement of technology, data is no longer observed at fixed moments of time, but continuously over a fixed interval in time or space (which we may identify with the interval $[0, 1]$). If this measurement is expected to exhibit a sudden drop or increase over this fixed interval, then an appropriate model for it could be a random element in an infinite dimensional space, such as the Skorohod space $D = D([0, 1])$ of cadlag functions on $[0, 1]$ (i.e. right-continuous functions with left limits).

In this talk, we examine the macroscopic limit (as time gets large) of the partial sum sequences associated to i.i.d. regularly varying elements with values in $D$. This limit is an interesting object in itself, which we call a "Dvalued stable Levy motion". Our methods were inspired by the construction of the classical stable Levy motion (in dimension $d$), and of its approximation by partial sums of i.i.d. regularly varying vectors, as presented in the monograph Resnick (2007). We extend these two results to the infinite-dimensional setting, using the concept of regular variation for random elements in $D$ introduced by de Haan and Lin (2001), and developed further by Hult and Lindskog (2005). The approximation result is an extension to functional convergence of a limit theorem obtained by Roueff and Soulier (2015).

This talk is based on joint work with Becem Saidani. 2018 — 19 Probabilistic Models in Quickest Change-Point Detection Aleksey Polunchenko (Binghamton University) Monday, 6th of May 2019 at in Malott 406 Probability Sequential (quickest) change-point detection is a branch of statistics concerned with the design and analysis of methods for rapid but reliable anomaly detection in "live" monitored random processes. The subject's areas of application are virtually unlimited, and include quality control, anomaly and failure detection, surveillance and security, finance, seismology, navigation, intrusion detection, boundary tracking---to name a few. This talk provides a brief overview of the field's state of the art. The overview includes a summary of the most popular probabilistic change-point models (minimax, Bayesian, generalized Bayesian) and a review of some of the "mainstream" change-point detection methods (CUSUM, Shiryaev's procedure and its variants). Certain open problems and challenges are discussed as well. Conformal embedding and percolation on the uniform triangulation Xin Sun (Columbia University) Monday, 29th of April 2019 at in Malott 406 Probability Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice has been achieved. Smirnorv’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for the critical percolation on uniform triangulations. Time permitting, I will also explain how this result fits into the larger picture of random planar maps and Liouville quantum gravity. Quantitative homogenization in a balanced random environment Xiaoqin Guo (University of Wisconsin-Madison) Monday, 22nd of April 2019 at in Malott 406 Probability Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice $\mathbb{Z}^d$. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). On the circle, Kahane's Gaussian Multiplicative Chaos and Circular Random Matrices match Reda Chhaibi (Institut de Mathématiques de Toulouse) Monday, 15th of April 2019 at in Malott 406 Probability In this talk, I would like to advertise an equality between two objects from very different areas of mathematical physics. This bridges the Gaussian Multiplicative Chaos, which plays an important role in certain conformal field theories, and a reference model in random matrices. On the one hand, in 1985, J.P Kahane introduced a random measure called the Gaussian Multiplicative Chaos (GMC). Morally, this is the measure whose Radon-Nikodym derivative w.r.t to Lebesgue is the exponential of a log correlated Gaussian field. In the cases of interest, this Gaussian field is a Schwartz distribution but not a function. As such, the construction of GMC needs to be done with care. In particular, in 2D, the GFF (Gaussian Free Field) is a random Schwartz distribution because of the logarithmic singularity of the Green kernel in 2D. Here we are interested in the 1D case on the circle. On the other hand, it is known since Verblunsky (1930s) that a probability measure on the circle is entirely determined by the coefficients appearing in the recurrence of orthogonal polynomials. Furthermore, Killip and Nenciu (2000s) have given a realization of the CBE, an important model in random matrices, thanks to random orthogonal polynomials of the circle. I will give the precise statement whose loose form is CBE = GMC. Joint work with J. Najnudel (Rescheduled to next semester) Moments of the Riemann Zeta Function on Short Intervals Louis Pierre-Arguin (The City University of New York) Monday, 8th of April 2019 at in Malott 406 Probability (Rescheduled to next semester) The moments of the Riemann zeta function on the interval [T,2T] on the critical line play a fundamental role in the distribution of the prime numbers. In this talk, we look at the moments of the Riemann zeta function on typical short intervals, but with length diverging with T. We will show that the moments exhibit a freezing phase transition up to a certain interval length akin to the transition seen in log-correlated processes. As a consequence we prove the leading order of the maximum of the zeta function on such short intervals. The results generalize a conjecture of Fyodorov & Keating and the related results of Arguin et al. and Najnudel on intervals of length one.

Joint work with F. Ouimet and M. Radziwill. Maximum and shape of 3D Ising interfaces Reza Gheissari (New York University) Monday, 25th of March 2019 at in Malott 406 Probability Dobrushin (1972) showed that the interface of a 3D Ising model with plus boundary conditions above the xy-plane and minus below is rigid (has $O(1)$ fluctuations) at low enough temperatures. We study the large deviations of this interface in a cube of side-length $n$, and obtain a law of large numbers for its maximum height $M_n$: for every inverse-temperature $\beta$ large enough, $M_n/\log n$ converges in probability to $h_\beta$ as $n\to\infty$.

We further show that on the large deviation event that the interface connects the origin to a height $H$, it consists of a 1D spine that behaves like a random walk in that it decomposes into asymptotically-stationary, weakly-dependent increments each of which has an exponential tail. These results generalize to every dimension $d\geq 3$.

Joint work with Eyal Lubetzky. Long jump random walks on finite nilpotent groups Yuwen Wang (Cornell University) Monday, 18th of March 2019 at in Malott 406 Probability In this talk, we consider a variant of random walks on finite groups. For this variant, at each step, we choose from a set of generators ("directions") uniformly, and an integer from a power law distribution ("speed") associated with the direction. We show that the mixing time of this walk of the same order of magnitude as the diameter of a pseudo-metric on the group that depends only on the speeds and generators. If time allows, I will talk about some of the challenges of computing this diameter. (Joint work with L. Saloff-Coste.) Double jump phase transition in a soliton cellular automaton John Pike (Bridgewater State University) Monday, 11th of March 2019 at in Malott 406 Probability I will discuss recent work with Lionel Levine and Hanbaek Lyu concerning the behavior of the soliton cellular automaton with random initial conditions. This CA is a discrete-time dynamical system which models the behavior of certain traveling wave packets arising in various areas of math and physics. After explaining the basics of the model, I will describe connections with a variety of combinatorial objects like pattern-avoiding permutations, Young tableaux, and Motzkin paths. I will then turn to the random setting where one can frame things in terms of probabilistic constructs like renewal processes, birth-and-death chains, Brownian motions, and Galton-Watson forests. Using these perspectives, I will present some limit theorems which establish a "double jump phase transition" for certain statistics of the system analogous to that found by Erdős and Rényi in their seminal study of random graphs. Generalized Ito-Nisio Theorem with applications to Levy processes and Levy driven SDEs Jan Rosinski (University of Tennessee Knoxville) Monday, 4th of March 2019 at in Malott 406 Probability The Ito-Nisio Theorem is a powerful tool that enables to deduce the uniform convergence from the pointwise convergence in Karkhunen-Loeve-type series expansions. Unfortunately, this tool fails for stronger than uniform modes of convergence, such as Lipschitz or p-variation convergence. On the other hand, the latter mode is natural in the study of processes with jumps. In this work we give a framework for and establish a generalization of the Ito-Nisio Theorem that covers also strong modes of convergence.

For applications to Levy processes, we find the optimal subspace of functions of finite phi-variation where all Levy processes without Gaussian component live and admit strongly converging series expansions. This yields an extension of the celebrated S.J. Taylor's theorem to Levy processes. We apply these results to Levy driven SDEs and infinitely divisible random fields.

This talk is based on a joint work with A. Basse-O'Connor and J. Hoffmann-Jorgensen. GOE Statistics for Levy Matrices Patrick Lopatto (Harvard University) Monday, 18th of February 2019 at in Malott 406 Probability We discuss recent work proving eigenvector delocalization and bulk eigenvalue universality for symmetric random matrices whose entries are independent $\alpha$-stable laws with $\alpha < 2$. Such distributions have infinite variance, and when $\alpha < 1$, infinite mean. In the latter case such matrices are conjectured to exhibit a sharp transition from a delocalized regime at low energy to a localized regime at high energy, like the infamous Anderson model in mathematical physics. These results are joint work with Amol Aggarwal and Horng-Tzer Yau. Self-Exciting Processes in Queues Andrew Daw (Cornell University) Monday, 11th of February 2019 at in Malott 406 Probability A point process is called "self-exciting" if the occurrence of an arrival increases the likelihood that additional arrivals occur soon after. These dynamics lead to temporal clustering of the arrivals and over-dispersion of the counting process. Often referred to as Hawkes processes, these processes have been the subject of recent study in a wide variety of applications, including finance, social media, seismology, and neurology. In this talk we explore the use of Hawkes processes in queueing theory and the impact of self-excitement on these models. Furthermore, we also investigate a model in which the self-exciting process responds to the queue, meaning that the arrival rate increases an entity arrives and then also decreases when the entity departs. In doing so, we both gain insight into the Hawkes process itself and connect self-exciting processes to other well known probability models. Survival probability for John and inner-uniform finite domains in integer lattices Laurent Saloff-Coste (Cornell University) Monday, 28th of January 2019 at in Malott 406 Probability This is a follow-up of may talk in the fall (but it will be presented independently, of course).

The problem is to understand the probability of survival up to time n in nice finite subsets of the square grid (in any fixed dimension).

How does this probability depend on the time and the starting point? Central Limit Theorems from the Location of Roots of Probability Generating Functions Marcus Michelen (University of Pennsylvania) Monday, 3rd of December 2018 at in Malott 406 Probability For a discrete random variable, what information can we find out from the roots of its probability generating function? We consider a sequence of random variables $X_n$ taking values between $0$ and $n$, and let $P_n(z)$ be its probability generating function. We show that if none of the complex zeros of the polynomials $P_n(z)$ are contained in a neighborhood of $1$ in the complex plane then a central limit theorem occurs, provided the variance of $X_n$ isn't subpolynomial in $n$. This result is sharp a sense that will be made precise, and thus disproves a conjecture of Pemantle and improves upon various results in the literature. This immediately improves a multivariate central limit theorem of Ghosh, Liggett and Pemantle, and has ramifications for certain variables that arise in graph theory contexts. Time permitting, we will describe other of our results connecting the location of the zeros of $P_n(z)$ and the distribution of the random variables $X_n$. This is based on joint work with Julian Sahasrabudhe. On the uniqueness/phase-coexistence threshold for the hard-core model in $\mathbb{Z}^d$ Prasad Tetali (Georgia Institute of Technology) Monday, 19th of November 2018 at in Malott 406 Probability It has long been conjectured that on the square lattice ($\mathbb{Z}^2$), the hard lattice gas model has a critical value $\lambda_c = 3.796…$ with the property: if $\lambda < \lambda_c$, then it exhibits uniqueness of phase, while if $\lambda > \lambda_c$ then there is phase coexistence – existence of multiple Gibbs measures.

The speaker will first review the basics of this model of independent interest in combinatorics, probability, statistical physics and theoretical computer science. Then he will give an update on the status of the problem on the square lattice, highlighting recent efforts that have rigorously established that $\lambda_c$ belongs to the interval [2.538, 5.3506], as well as mentioning related open problems of interest. Fluctuation lower bounds in planar random growth models Erik Bates (Stanford University) Monday, 12th of November 2018 at in Malott 406 Probability Even after years of study on random growth models, such as first- and last-passage percolation and directed polymers, much remains mysterious or out of reach technically. In particular, beyond the fundamental shape theorems guaranteeing linear growth rates for the passage times/free energy, there are sublinear fluctuations whose asymptotics are not established. Concerning the planar versions of these models, KPZ universality predicts order $n^{1/3}$ fluctuations. In this talk, I will discuss new methods for obtaining a lower bound of $\sqrt{\log n}$, which (in a frustrating state of affairs) is the best known result in general. This is joint work with Sourav Chatterjee. Random sup-measures and long range dependence Zaoli Chen (Cornell University) Monday, 5th of November 2018 at in Malott 406 Probability In this talk, I will talk about a useful tool in extremal value theory. Extremes of a stationary sequence can be fitted into the framework of random sup-measures, which was pointed out by O'Brien et al in the 1980s. I will explain the random sup-measures, and see their applications in the heavy-tailed settings. A specific random sup-measure arising from a heavy-tailed stationary process with long-range dependence will be explored. I will also explain how the dependence structures are reflected in the limiting theorem. Detection of Anomalous Path in a Noisy Network Shirshendu Chatterjee (The City College of New York) Monday, 29th of October 2018 at in Mallot 406 Probability Consider a two dimensional finite graph having side length $O(n)$. Each vertex of the graph is associated with a random variable, and these are assumed to be independent. In this setting, we will consider the following hypothesis testing problem. Under the null, all the random variables have common distribution $N(0, 1)$, while under the alternative, there is an unknown path (with unknown initial vertex) having $O(n)$ edges (e.g. a "left to right crossing") along which the associated random variables have distribution $N(\mu n, 1)$ for some $\mu n > 0$, and the random variables away from the path have distribution $N(0, 1)$. We will describe the values of the mean shift $\mu n$ for which one can reliably detect (in the minimax sense) the presence of the anomalous path, and for which it is impossible to detect. Diffusion-Limited Annihilating Systems Matt Junge (Duke University) Monday, 22nd of October 2018 at in Mallot 406 Probability We study a two-type annihilating system in which particles are placed with equal density on the integer lattice. Particles perform simple random walk and annihilate when they contact a particle of different type. The density of particles at the origin was known to exhibit anomalous behavior in low-dimension when particles have equal speeds. Describing the setting with asymmetric speeds has been open for over 20 years. We prove a lower bound that matches physicists' conjectures and discuss partial progress towards an upper bound. Joint with Michael Damron, Hanbaek Lyu, Tobias Johnson, and David Sivakoff. Central limit theorems by Jack generating functions Jiaoyang Huang (Harvard University) Monday, 15th of October 2018 at in Mallot 406 Probability One classical approach for the proof of the central limit theorem is via the characteristic functions, which is the Fourier transform of the probability density. In this talk, I will discuss an approach to study the global fluctuations of random N-particle systems using the Jack polynomials, which can be viewed as the Fourier transform on certain locally compact groups. I will first introduce the Jack polynomials, and give some examples of random N-particle systems related Jack polynomials. Then I will introduce the Jack generating functions, which generalize the Schur generating functions introduced by Bufetov and Vadim. Finally, I will explain how to extract information of the measure by applying certain operators on the Jack generating functions, and survey some previous works using this idea. Lower-tail large deviations of the KPZ equation Li Cheng Tsai (Columbia University) Monday, 1st of October 2018 at in Mallot 406 Probability Regarding time as a scaling parameter, we prove the one-point, lower tail Large Deviation Principle (LDP) of the KPZ equation, with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from [Borodin Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform. Algorithmic thresholds for tensor principle component analysis Aukosh Jagannath (Harvard University) Monday, 24th of September 2018 at in Mallot 406 Probability Consider the problem of recovering a rank 1 tensor of order k that has been subject to Gaussian noise. The log-likelihood for this problem is highly non-convex. It is information theoretically possible to recover the tensor with a finite number of samples via maximum likelihood estimation, however, it is expected that one needs a polynomially diverging number of samples to efficiently recover it. What is the cause of this large statistical–to–algorithmic gap? To study this question, we investigate the thresholds for efficient recovery for a simple family of algorithms, Langevin dynamics and gradient descent. We view this problem as a member of a broader class of problems which correspond to recovering a signal from a non-linear observation that has been perturbed by isotropic Gaussian noise. We propose a mechanism for success/failure of recovery of such algorithms in terms of the strength of the signal on the high entropy region of the initialization. Joint work with G. Ben Arous (NYU) and R. Gheissari (NYU). Lower bounds for fluctuations in first-passage percolation Michael Damron (Georgia Institute of Technology) Monday, 17th of September 2018 at in Malott 406 Probability In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice $\mathbb{Z}^d$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than $1$, but the best lower bounds available are (only in two dimensions) of order $\log |x|$. This result was found in the '90s and there has not been any improvement since. In this talk, we discuss the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\log |x|)^{1/2}$, and similar statements for FPP in thin cylinders. Such a statement has been proved for special edge-weight distributions by Pemantle-Peres ('95) and Chatterjee ('17). In ongoing work with J. Hanson, C. Houdré, and C. Xu, we aim to extend these bounds to general edge-weight distributions. I will explain some of the methods we are using, including an old and elementary "small ball" probability result for functions on the hypercube. Doob's transform and finite Markov chains Laurent Saloff Coste (Cornell University) Monday, 27th of August 2018 at in Mallot 406 Probability In this seminar, I will explain how the fundamental concept of Doob's transform allow us to reduce some problems regarding absorbing finite Markov chains to problems regarding ergodic finite Markov chains for which many tools are available. The talk will be elementary even though I will explain the relevance of some more sophisticated techniques. 2017 — 18 The Archimedean limit of random sorting networks Duncan Dauvergne (University of Toronto) Monday, 7th of May 2018 at in Mallot 406 Probability A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of $S_n$ generated by adjacent transpositions. An $n$-element uniform random sorting network displays many striking global properties as $n$ approaches infinity. For example, scaled trajectories of the elements $1, 2, \dotsc, n$ converge to sine curves and the $1/2$-way permutation matrix measure converges to the projected surface area measure of the $2$-sphere.

In this talk, I will discuss how the local structure of random sorting networks can be used to find a global limit, proving these statements and more. Learning Discrete Determinantal Point Processes Victor-Emmanuel Brunel (Massachusetts Institute of Technology) Monday, 16th of April 2018 at in Mallot 406 Probability Discrete Determinantal Point Processes (DPPs) are a class of probabilistic models that describe repulsive interactions between items. Unlike for most graphical models (e.g., Markov Random Fields, including the ferromagnetic Ising model), many simple yet useful operations, such as conditioning and marginalizing, are tractable with DPPs. This is why they have gained a lot of popularity in numerous applications in the past few years. In this presentation, I will talk about learning the parameter of a DPP, given independent observed copies. The first approach that I will describe is the maximum likelihood estimation. I will mention its asymptotic guarantees and I will explain some computational challenges that it poses. The second approach is based on a method of moments, which relies on a problem called the principal minor assignment problem: Given the list of all principal minors of some matrix, how to recover that matrix in a computationally efficient way?

This talk is based on two joint works with Ankur Moitra, Philippe Rigollet and John Urschell. Stochastic quantization of gauge theories Hao Shen (Columbia University) Tuesday, 10th of April 2018 at in 203 Malott Probability "Stochastic quantization" refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the $\Phi^4_3$ equation.

In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables. Formation of large-scale random structure by competitive erosion Sourav Sarkar (University of California Berkeley) Monday, 26th of March 2018 at in Malott 406 Probability Competitive erosion models a random interface sustained in equilibrium by equal and opposite pressures on each side of the interface. Here we study the following one dimensional version. Begin with all sites of $\mathbb{Z}$ uncolored. A blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after n blue and n red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$ . The resulting random color configuration has a certain fractal nature which after scaling by $n^{1/4}$ and taking a limit, has an explicit description in terms of alternating extrema of Brownian motions.

This is a joint work with Shirshendu Ganguly and Lionel Levine. Sofic and percolative entropies of Gibbs measures on regular infinite trees Moumanti Podder (Georgia Institute of Technology) Monday, 19th of March 2018 at in Malott 406 Probability Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the percolative entropy $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. When it is known to exist, the limit of $|V_n|^{-1}H(\mu_n)$ is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. Algebraic constructions of Markov duality functions Jeffrey Kuan (Columbia University) Monday, 12th of March 2018 at in Malott 406 Probability Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues. On the extreme values of the Riemann zeta function on random intervals of the critical line Joseph Najnudel (University of Cincinnati) Friday, 9th of March 2018 at in Malott 205 Probability In this talk, we study the maximum of the real and the imaginary part of the logarithm of the Riemann zeta function on random intervals of the critical line. We sketch a proof of the fact that if these intervals are centered at 1/2 + i UT, U being uniformly distributed on [0,1], then the maximum, divided by log log T, tends to 1 in probability. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating. Local statistics of Dyson Brownian motion Benjamin Landon (Harvard University) Monday, 26th of February 2018 at in Malott 406 Probability Dyson Brownian motion is a stochastic eigenvalue dynamics and the basis of the dynamical approach to random matrix theory. We review recent results on the local statistics of Dyson Brownian motion, as well as how these results are applied to prove the universality of general random matrix ensembles. Based on joint work with Z. Che, J. Huang, P. Sosoe and H.-T. Yau. Poincare inequality on manifolds with ends Satoshi Ishiwata (University of Tsukuba) Monday, 12th of February 2018 at in Malott 406 Probability Recently, heat kernel estimates on manifolds with ends has been obtained by Grigor'yan and Saloff-Coste, and the speaker. In view of these results, it is natural to ask whether the Poincare inequality holds or not on these manifolds. In this talk we give an answer of this question. This talk is based on a joint work with Alexander Grigor'yan and Laurent Saloff-Coste. The evolutionary dynamics of incubation periods Bertrand Ottino-Loffler (Cornell University) Monday, 5th of February 2018 at in Malott 406 Probability The incubation period for typhoid, strep, leukemia, bladder cancer, and many other diseases follows a right-skewed, approximately lognormal distribution. Although this pattern was discovered over sixty years ago, it remains an open question to explain its ubiquity. Here we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on biological heterogeneity, our results hold even for homogeneous systems. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease. On set-valued Markov intertwinings Laurent Miclo (University of Toulouse) Monday, 29th of January 2018 at in Malott 406 Probability A Markov intertwining relation between two Markov processes $X$ and $Y$ is a weak similitude relation $G\Lambda=\Lambda L$ between their generators $L$ and $G$, where $\Lambda$ is a transition kernel between the underlying state spaces.

This notion is an important tool to deduce quantitative estimates on the speed of convergence to equilibrium of $X$ via strong stationary times when $Y$ is absorbed, as shown by the theory of Diaconis and Fill for finite state spaces.

In this talk we will only consider processes $Y$ taking as values some subsets of the state space of $X$.

Our goal is to present extensions of the above method to elliptic diffusion processes on differentiable manifolds, via stochastic modifications of mean curvature flows.

We will see that Pitman's theorem about the intertwining relation between the Brownian motion and the Bessel-3 process is curiously ubiquitous in this approach.

It even serves as an inspiring guide to construct couplings associated to finite Markov intertwining relations via random mappings, in the spirit of the coupling-from-the-past algorithm of Propp and Wilson and of the evolving sets of Morris and Peres. The boundary in first-passage percolation Jack Hanson (The City College of New York) Monday, 27th of November 2017 at in Malott 406 Probability First-passage percolation can be thought of as inducing a growth process on the graph $\mathbb{Z}^d$. The "infected region" $B(t)$ at time $t$ is known to have diameter linearly growing in $t$. We describe work showing that, if the i.i.d. edge weights generating the model satisfy a weak moment condition, the cardinality of the boundary of $B(t)$ is of order $t^{d-1}$ for most times. For heavy-tailed variables, the boundary is larger, due to the presence of small holes, but the exterior boundary is still of order $t^{d-1}$. Under a further (but standard) unproven "curvature" assumption, we show that the boundary has cardinality at most $(\log t)^C t^{d-1}$ for all large $t$. Wandering exponent for random walk in a dynamic beta environment Firas Rassoul-Agha (University of Utah) Monday, 20th of November 2017 at in Malott 406 Probability Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The transform utilizes a harmonic function defined by a Busemann-type limit. The Doob-transformed environment is correlated in time, i.i.d. in space, and under its averaged distribution the transformed walk obeys the wandering exponent $2/3$ that agrees with Kardar-Parisi-Zhang universality. This is a joint work with Marton Balazs and Timo Seppalainen. Polluted bootstrap percolation Alexander E. Holroyd (University of Washington) Monday, 13th of November 2017 at in Malott 406 Probability Bootstrap percolation is a fundamental cellular automaton model for nucleation. Despite its simplicity, the model holds many surprises. I'll focus on how growth from sparse random seeds is affected by sparse random impurities in the medium. The answer will involve using recent oriented surface technology to construct a stegosaurus.

Based on joint work with Janko Gravner and David Sivakoff. Kernel-based methods for bandit convex optimization Sébastien Bubeck (Microsoft Research) Monday, 6th of November 2017 at in Gates 114 Probability A lot of progress has been made in recent years on extending classical multi-armed bandit strategies to very large set of actions. A particularly challenging setting is the one where the action set is continuous and the underlying cost function is convex, this is the so-called bandit convex optimization (BCO) problem. I will tell the story of BCO and explain some of the new ideas that we recently developed to solve it. I will focus on three new ideas from our recent work arXiv:1607.03084 with Yin Tat Lee and Ronen Eldan: (i) a new connection between kernel methods and the popular multiplicative weights strategy; (ii) a new connection between kernel methods and one of Erdős’ favorite mathematical object, the Bernoulli convolution, and (iii) a new adaptive (and increasing!) learning rate for multiplicative weights. These ideas could be of broader interest in learning/algorithm’s design. The reset time of Internal DLA Vittoria Silvestri (Cambridge University) Monday, 30th of October 2017 at in Malott 406 Probability Consider Internal DLA on cylinder graphs of the form GxZ. How does a large cluster typically look like? How long does it take for the process to forget its initial profile? In this talk I will address these questions, explaining how the answer depends on the mixing properties of the base graph G. Joint work with Lionel Levine. Discrete rough paths and limit theorems Samy Tindel (Purdue University) Monday, 23rd of October 2017 at in Malott 406 Probability In this talk I will focus on a series of results concerning Breuer-Major type theorems, p-variation limits, as well as Itô type formulas in law for Gaussian processes. This line of research has been quite active in the recent past in the stochastic analysis community. Most of the techniques involve integration by parts, Stein’s method, and other Malliavin calculus tools. This yields a series of limitations on the nature of the results, as well as the dimension of the Gaussian process at stake. My aim is to show how those questions can be handled in a more natural way thanks to rough path type techniques.

More specifically I will show how to transfer limits taken on a Gaussian signature to limits involving controlled processes, by means of the typical expansions of the rough paths theory. Applications of this rather simple trick include the aforementioned p-variations and Itô type formulas, as well as central limit theorems for numerical schemes.

I will try to introduce most of my notations carefully, and keep the needed rough path background to a minimum level. The talk is based on joint works with Yanghui Liu (Purdue). Extreme Level Sets of Branching Brownian Motion Lisa Hartung (New York University) Monday, 16th of October 2017 at in Malott 406 Probability We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor) Spectral reduction of non-self-adjoint self-similar semigroups Pierre Patie (Cornell University) Monday, 2nd of October 2017 at in Malott 406 Probability The aim of this talk is to provide a thorough spectral analysis of the class, denoted by K, of markovian self-similar Co-semigroup on L2. This class of operators, which are associated to non-self-adjoint and non-local generators, appears in various frameworks such as the study of growth-fragmentation equations, random planar maps and stable processes.

We resort to the concept of intertwining (a weak version of similarity) which turns out to be a natural and powerful technique to develop the spectral reduction of non-normal operators. We start by providing a (brief) historical account on the connection between spectral theory of linear operators and intertwining, including the notion of spectral operators introduced by Dunford. We proceed by showing that the intertwining orbit of each element in K is K itself, meaning that any two elements in K are intertwined with a closed linear (non-necessarily positive, neither bounded) operator. Relying on these commutation relationships, we characterize, for each semigroup in K, its spectrum which can be either point, continuous or residual. We end the talk by describing the associated (weak) eigenfunctions as (weak) Fourier kernels, the spectral reduction operator and its domain. (Joint work with M. Savov) Sandpiles as dynamical systems Lionel Levine (Cornell University) Monday, 18th of September 2017 at in Malott 406 ProbabilityDynamical Systems In this talk I’ll take a dynamical view on certain questions in probability.

Perhaps the simplest question about any dynamical system is whether it converges to a fixed point. A classic example in probability is whether a branching process dies out.

For a spatially infinite system, we can ask about *local* fixation: does every bounded subset eventually stop changing? A classic example is whether a random walk is transient or recurrent.

In the case of sandpile models (I’ll define them precisely in the talk) we can ask whether an avalanche will stop or go on forever. In arXiv:1508.00161 Hannah Cairns proved for 3-dimensional abelian sandpiles that this question is algorithmically undecidable: it is as hard as the halting problem! But this infinite unclimbable peak is surrounded by appealing finite peaks: What about 2 dimensions? What if the initial configuration of sand is i.i.d.? Deciding local fixation of i.i.d. abelian sandpiles is still open, even on the square grid $\mathbb{Z}^2$. I’ll tell you about the "mod 1 harmonic functions" Bob Hough and Daniel Jerison and I used to prove in arXiv:1707.06081, but the actual value of the critical density is still unknown even on $\mathbb{Z}$.

This is a joint talk with the Dynamical Systems Seminar. Trace reconstruction for the deletion channel Yuval Peres (Microsoft Research) Monday, 11th of September 2017 at in Malott 406 Probability In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability q, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability?

The best lower bound known is linear in $n$. Until recently, the best upper bound was exponential in the square root of $n$. We improve the square root to a cube root using statistics of individual output bits and some complex analysis; this bound is sharp for reconstruction algorithms that only use this statistical information. (Similar results were obtained independently and concurrently by De, O’Donnell and Servedio). If the string $x$ is random and $q < 1/2$, we can show a subpolynomial number of traces suffices by comparison to a biased random walk. (Joint works with Fedor Nazarov, STOC 2017 and with Alex Zhai, FOCS 2017). 2016 — 17 Airy scaling of random Hermite functions at the caustic Boris Hanin (Massachusetts Institute of Technology) Thursday, 11th of May 2017 at in Malott 406 Probability Random Hermite functions are eigenfunctions of fixed energy for the isotropic harmonic oscillator in $\mathbb{R}^d$. They therefore transition from highly oscillatory to exponentially damped at the caustic, the analog of the edge of the spectrum in random matrix theory. The purpose of this talk is to explain how the Airy kernel and its relatives appear in the scaling limit of random Hermite functions around the caustic as Planck's constant h goes to 0. Rather than being the kernel of the determinantal process, the Airy kernel will appear as the covariance function of a limiting Gaussian field.

This is joint work with Steve Zelditch and Peng Zhou. Analyzing Markov chains using belief propagation Eric Vigoda (Georgia Institute of Technology) Monday, 8th of May 2017 at in Malott 406 Probability For counting weighted independent sets weighted by a parameter $\lambda$ (known as the hard-core model) there is a beautiful connection between statistical physics phase transitions on infinite, regular trees and the computational complexity of approximate counting on graphs of maximum degree $D$. For $\lambda$ below the critical point there is an elegant algorithm devised by Weitz (2006). The drawback of his algorithm is that the running time is exponential in $\log D$. In this talk we’ll describe recent work which shows $O(n \log n)$ mixing time of the single-site Markov chain when the girth > 6 and $D$ is at least a sufficiently large constant. Our proof utilizes BP (belief propagation) to design a distance function in our coupling analysis. This is joint work with C. Efthymiou, T. Hayes, D. Stefankovic, and Y. Yin, see: arXiv:1604.01422 Probability, dynamics, and the geometry of numbers Jayadev Athreya (University of Washington) Monday, 1st of May 2017 at in Malott 406 Probability Motivated by a classical problem in probabilistic number theory posed by Erdős-Szüsz-Turán, we describe how dynamics, probability, and the geometry of numbers can come together when studying Diophantine approximation. Parts of this talk are joint work with Anish Ghosh, and other parts are joint work with G. Margulis. Information-theoretic bounds and phase transitions in community detection and high-dimensional clustering Cristopher Moore (Santa Fe Institute) Monday, 24th of April 2017 at in Rhodes 253 Probability Over the past decade, a rich interaction has grown up between statistical physics and statistical inference. In particular, physics often predicts phase transitions where, if a data set becomes too sparse or too noisy, it suddenly becomes impossible to find its underlying structure, or even to distinguish it from a "null model" with no structure at all. For instance, in the community detection problem in graphs, there is a transition below which the vertices cannot be labeled better than chance. Similarly, in the clustering problem in Gaussian mixture models, it becomes impossible to find the clusters, or even to tell whether clusters exist, i.e., to distinguish the data from a single Gaussian.

Many of the physics predictions have been now made rigorous, but there is still a lively interchange between these fields, both about these transitions and about asymptotically optimal algorithms. In particular, while efficient message-passing and spectral algorithms are known that succeed down to the so-called Kesten-Stigum bound, in a number of problems there is a conjectured "possible but hard" regime where inference is information-theoretically possible, but conjectured to be computationally hard. I will give a friendly introduction to this field, and present some recent results along these lines.

This is joint work with Jess Banks, Joe Neeman, Praneeth Netrapalli, and Jiaming Xu. Limit theorems for hyperbolic random geometric graph model Takashi Owada (Purdue University) Monday, 17th of April 2017 at in Malott 406 Probability We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchical structure. From this viewpoint, we discuss the asymptotic behavior of subtree counts. It then turns out that the spatial distribution of subtrees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, subtrees are asymptotically scattered near the center of the Poincare ball. On the contrary, if the space becomes "more hyperbolic" (i.e., more negatively curved), the distribution of trees is asymptotically determined by those concentrated near the boundary of the Poincare ball.

This is joint work with Yogeshwaran D. at Indian Statistical Institute. Large deviations for the volume of $k$-nearest neighbor balls Takashi Owada (Purdue University) Monday, 17th of April 2017 at in Mallot 406 Probability We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchical structure. From this viewpoint, we discuss the asymptotic behavior of subtree counts. It then turns out that the spatial distribution of subtrees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, subtrees are asymptotically scattered near the center of the Poincare ball. On the contrary, if the space becomes "more hyperbolic" (i.e., more negatively curved), the distribution of trees is asymptotically determined by those concentrated near the boundary of the Poincare ball.

This is joint work with Yogeshwaran D. at Indian Statistical Institute. The travel time to infinity in percolation Michael Damron (Georgia Institute of Technology) Monday, 10th of April 2017 at in Malott 406 Probability On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the edges with common distribution $F$. For which distributions $F$ is there an infinite self-avoiding path with finite total weight? This question arises in first-passage percolation, the study of the random metric space $\mathbb{Z}^2$ with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when $F(0) < 1/2$ (there are only finite paths of zero-weight edges), and there is one when $F(0) > 1/2$ (there is an infinite path of zero-weight edges). The critical case, $F(0) = 1/2$, is considerably more difficult due to the presence of finite paths of zero-weight edges on all scales. I will discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient conditions on $F$ for the existence of an infinite finite-weight path. The methods involve comparing the model to another one, invasion percolation, and showing that geodesics in first-passage percolation have the same first order travel time as optimal paths in an embedded invasion cluster. Oracle-efficient online learning and applications to auction design Nika Haghtalab (Carnegie Mellon University) Monday, 27th of March 2017 at in Malott 406 Probability We consider the fundamental problem of learning from expert advice, a.k.a. online no-regret learning, where we have access to an offline optimization oracle that can be used to compute, in constant time, the best performing expert at any point in time. We consider the design of no-regret algorithms that are computationally efficient using such an oracle.

We present structural properties under which we show oracle-efficient no-regret algorithms exist, even when the set of experts is exponentially large in a succinct representation of the problem. Our algorithm is a generalization of the Follow-The-Perturbed-Leader algorithm of Kalai and Vempala that at every step follows the best-performing expert subject to some perturbations. Our design uses a shared source of randomness across all experts that can be efficiently implemented by using an oracle on a random modification of the history of the play at every time step.

Our second main contribution is showing that the structural properties required for our oracle-efficient online algorithm are present in a large class of problems. As an example, we discuss the applications of our oracle-efficient learning results to the adaptive optimization of a large class of auctions, including (1) VCG auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) Myerson-like auctions for single-item settings. An infinite-dimensional Skorokhod map and continuous parameter priorities Haya Kaspi (Technion -- Israel Institute of Technology) Monday, 20th of March 2017 at in Malott 406 Probability The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this lecture I'll introduce a measure-valued analog of this map that transforms each element of a certain class of càdlàg paths that take values in the space of signed measures on the positive half line to a càdlàg path that takes values in the space of non-negative measures on that space. This is done in a way that for each point $x > 0$, and a signed measure valued process $(m_t)$ the path $t \mapsto m_t[0, x]$ is transformed via the classical Skorokhod map on the half-line, and the regulating functions for different $x > 0$ are coupled. We show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the EDF- earliest-deadline-first, the SJF- shortest-job-first and the SRPT- shortest-remaining-processing-time scheduling policies. Concentrating in this talk on the EDF, I’ll show how the map provides a framework within which one forms fluid model equations, proves uniqueness of solutions to these equations and establishes convergence of scaled state processes to the fluid model. In particular, for this model, our approach leads to new convergence results in time-inhomogeneous settings, which appear to fall outside the scope of existing approaches.

Joint work with Rami Atar, Anup Biswas and Kavita Ramanan. Convex low-rank matrix optimization with optimal storage Madeleine Udell (Cornell University) Monday, 13th of March 2017 at in Malott 406 Probability This talk concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modifies a standard convex optimization scheme, the conditional gradient method, to store only a small randomized sketch of the matrix variable. After the optimization terminates, the algorithm extracts a low-rank approximation of the solution from the sketch. In contrast to nonconvex heuristics, the guarantees for SketchyCGM do not rely on statistical models for the problem data. Numerical work demonstrates the benefits of SketchyCGM over heuristics. Probabilistic symmetries and random locations Yi Shen (University of Waterloo) Monday, 6th of March 2017 at in Malott 406 Probability In this talk we briefly discuss different types of symmetries in probability, including stationarity, stationarity of the increments, isotropy, self-similarity and their combinations. Each of these symmetries can be expressed as the invariance of the distribution of the stochastic processes with respect to certain action. In particular, we consider the random locations of stochastic processes, such as the hitting times or the location of the path supremum over a fixed interval. On one hand, we see how the probabilistic symmetries can imply the properties of the distributions of these random locations; on the other hand, we also discuss how the distributions of the random locations can be used to characterize the probabilistic symmetries. We then proceed to propose a unified framework to study the random locations for stochastic processes exhibiting probabilistic symmetries. Optimal concentration of information for log-concave distributions Mokshay Madiman (University of Delaware) Monday, 27th of February 2017 at in Malott 406 Probability It was shown by Bobkov and the speaker that for a random vector $X$ in $\mathbb{R}^n$ drawn from a log-concave density $e^{-V}$, the information content per coordinate, namely $V(X)/n$, is highly concentrated about its mean. Their argument was nontrivial, involving the localization technique, and also gave suboptimal exponents, but it was sufficient to demonstrate that high-dimensional log-concave measures are in a sense close to uniform distributions on the annulus between 2 nested convex sets (generalizing the well known fact that the standard Gaussian measure is concentrated on a thin spherical annulus). We will present recent work that obtains an optimal concentration bound in this setting (optimal even in the constant terms, not just the exponent), using very simple techniques, and outline the proof. Applications that motivated the development of these results include high-dimensional convex geometry and random matrix theory, and we will outline these applications. Based on (multiple) joint works with Sergey Bobkov, Matthieu Fradelizi, and Liyao Wang. Convergence of the self-avoiding walk on random quadrangulations to SLE Ewain Gwynne (Massachusetts Institute of Technology) Wednesday, 15th of February 2017 at in Malott 205 Probability We prove that a uniform infinite quadrangulation of the half-plane decorated by a chordal self-avoiding walk (SAW) converges in the scaling limit to SLE$_{8/3}$ on an independent $\sqrt{8/3}$-Liouville quantum gravity surface, which can equivalently be described as the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. The proof of the scaling limit result uses only the theory of random planar maps and does not make direct use of SLE or LQG. Based on joint work with Jason Miller.
arXiv:1608.00956 Coeulerian graphs and random matrices Shaked Koplewitz (Yale University) Monday, 6th of February 2017 at in Malott 406 Probability A directed graph is called Eulerian if it has an Eulerian circuit, which is equivalent to requiring that the kernel of its Laplacian matrix is minimal. We call a directed graph coEulerian if the cokernel of its Laplacian matrix is maximal. While it is easy to see that a random graph is almost never Eulerian, calculating the probability that it is coEulerian is considerably harder. We sketch a proof that the limit of this probability is bounded by a constant around 0.43, and talk about the obstacles to proving that the limit is exactly this constant. In the last part of the talk, we talk about results and conjectures on the probability that a random rectangular matrix is surjective, and their implications for random graph theory. Loop-erased random surface Kyle Parsons (The Ohio State University) Monday, 5th of December 2016 at in Malott 406 Probability Loop-erased random walk and its scaling limit, Schramm-Loewner evolution, have found numerous applications in mathematics and physics. We present a 2-dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2-dimensional spanning tree and declaring that LERS should have the same relation to these 2-trees as LERW has to ordinary spanning trees. Furthermore we present numerical evidence that the growth rate for LERS on a $\delta$-fine grid as $\delta \to 0$ is approximately 2.53. This suggests the possibility of a fractal limiting object for LERS analogous to SLE for LERW. Cut-off for a class of hyperplane arrangement walks Evita Nestoridi (Princeton University) Monday, 28th of November 2016 at in Malott 406 Probability Consider a real hyperplane arrangement and let C denote the collection of the occuring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on C which is a generalization of some card shuffling models used in computer science, biology and card games: the famous Tsetlin library used in dynamic file maintenance and cache maintenance and the riffle shuffles are two important examples of hyperplane walks. A strong stationary argument gives the upper bound for the separation distance mixing time for this Markov chain. A coupon collector argument will lead to the lower bound, which is discussed for the first time. I will try to explain both the geometric and the probabilistic techniques used in the problem. A reverse Minkowski theorem Noah Stephens-Davidowitz (New York University) Monday, 21st of November 2016 at in Malott 406 Probability A classical question in the geometry of numbers asks how many lattice points lie in some ball around the origin. Minkowski's celebrated theorem gives us a tight lower bound for this number that depends only on the determinant of the lattice. (One can think of the determinant as the limit of the density of lattice points inside large balls. So, Minkowski's theorem gives a tight lower bound on a lattice's "local density" based on its "global density"). We resolve a conjecture due to Dadush that gives a nearly tight converse to Minkowski's theorem—an upper bound on the number of lattice points in a ball that depends only on the determinants of sublattices. This theorem has numerous applications, from complexity theory, to the geometry of numbers, to the behavior of Brownian motion on flat tori.

Based on joint work with Oded Regev. The Schelling model Nina Holden (Massachusetts Institute of Technology) Monday, 14th of November 2016 at in Malott 406 Probability The Schelling model of segregation describes how a population of multiple types self-organize to form homogeneous clusters of one type. It was introduced by Nobel price winning economist Schelling in 1969, and is one of the earliest and most influential interacting-agent models studied by economists. We present the first mathematical description of the dynamical scaling limit of this model as the interaction length between the agents tends to infinity. In the one-dimensional case we describe the scaling limit of the limiting clusters obtained at time infinity. Geometry of random sorting networks Mustazee Rahman (Massachusetts Institute of Technology) Monday, 7th of November 2016 at in Malott 406 Probability Sorting networks are shortest paths between the identity and the reverse permutation in the Cayley graph of the symmetric group generated by adjacent transpositions. Some remarkable conjectures have been made on the geometric behavior of uniform random sorting networks. For example, the half way permutation appears to be the projection of spherical measure onto the disk. I will explain these conjectures, its relation to the limit theory of permutations and some recent progress supporting these conjectures in joint work with Balint Virag and Mate Vizer. Current fluctuations of the stationary ASEP and six-vertex model Amol Aggarwal (Harvard University) Monday, 31st of October 2016 at in Malott 406 Probability We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them. Excited random walks in Markovian cookie environments on Elena Kosygina (City University of New York, Baruch College) Monday, 24th of October 2016 at in Malott 406 Probability We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $w(x,n)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $n$ to $x$. The collection $\{w(x,n): x, n \text{ are integers, } n \text{ is non-negative}\}$, is sometimes called a "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probability according to the "flavor" of the cookie eaten. Assume that the cookie stacks are i.i.d. and that the cookie "flavors" at each stack $w(x,n)$, $n=0,1,\ldots$ follow a finite state Markov chain in $n$. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the random walk time. We discuss recurrence/transience, ballisticity, and limit theorems for such walks. This is a joint work with Jonathon Peterson, Purdue University. Exact recovery of chaotic systems from highly corrupted data Giang Tran (University of Texas at Austin) Monday, 17th of October 2016 at in Malott 406 Probability Learning the governing equations in dynamical systems from time-varying measurements is of great interest across different scientific fields. When the underlying system exhibits chaotic behaviors, such as sensitivity to initial conditions, it is crucial to recover the governing equations with high precision. In this work, we bring together connections between compressed sensing, splitting optimization methods, sparse representations of the governing equations, and the statistical properties of chaotic systems -- to provide exact recovery guarantees for classes of dynamical systems. As our main theoretical result, we show that if a system is sufficiently ergodic, thus its data satisfies a strong central limit theorem (as is known to hold for chaotic Lorenz systems), then the governing equations can be exactly recovered as the solution to an L1 minimization problem -- even if a large percentage of the data is corrupted by outliers. Numerically, we apply the alternating minimization method to solve the corresponding constrained optimization problem. We illustrate the power, generality, and efficiency of our model through several examples of 3D chaotic systems and higher dimensional hyper-chaotic systems. Random zero sets under repeated differentiation of an analytic function Sneha Subramanian (Georgia Institute of Technology) Monday, 3rd of October 2016 at in Malott 406 Probability For a random (complex) entire function, what can we say about the behavior of the zero set of its $N$th derivative, as $N$ goes to infinity? In this talk, we shall discuss the result of repeatedly differentiating a certain class of random entire functions whose zeros are the points of a Poisson process of intensity 1 on the real line. Based on joint work with Robin Pemantle. Time-changed extremal process as a random sup measure Gennady Samorodnitsky (Cornell University) Monday, 26th of September 2016 at in Malott 406 Probability A functional limit theorem for the partial maxima of a long memory stable sequence produces a limiting process that can be described as a beta-power time change in the classical Frechet extremal process, for beta in a subinterval of the unit interval. Any such power time change in the extremal process for 0<\beta<1 produces a process with stationary max-increments. This deceptively simple time change hides the much more delicate structure of the resulting process as a self-affine random sup measure. We uncover this structure and show that in a certain range of the parameters this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. These results open a way to construct a whole new class of self-similar Frechet processes with stationary max-increments. Non-fixation for activated random walks with small particle density Riddhipratim Basu (Stanford University) Monday, 19th of September 2016 at in Malott 406 Probability Activated Random Walk (ARW) is an interacting particle system that approximates the Stochastic Sandpile Model (SSM), a paradigm example of a model of self-organized criticality. On $\mathbb{Z}$, one starts with a mass density $\mu$ of initially active particles each of which performs a continuous time nearest neighbour symmetric random walk at rate one and falls asleep at rate $\lambda>0$. Sleepy particles become active on coming in contact with active particles. I shall describe a recent joint work with Shirshendu Ganguly and Christopher Hoffman where we use a novel renormalized variant of the Diaconis-Fulton construction of the process and the associated Abelian property to show that even at arbitrarily small positive desnity of particles, the system does not fixate provided the sleep rate $\lambda$ is sufficiently small. This answers positively two open questions from Rolla and Sidoravicius (Invent. Math., 2012). Random walks: Volume growth and beyond Laurent Saloff-Coste (Cornell University) Monday, 12th of September 2016 at in Malott 406 Probability In this talk, I will describe a somewhat new approach to basic results about random walks on groups, questions about random walks driven by spread-out measures, and examples that demonstrate the rich interactions between group structure and the behavior of random walks. Random matrices, the Hadamard product and the free convolutions Arijit Chakrabarty (Indian Statistical Institute, Delhi Centre) Tuesday, 6th of September 2016 at in Malott 207 Probability Random matrices whose entries come from a stationary Gaussian process are studied. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behaviour of the eigenvalues in a completely different way.

The random matrix results obtained are used to understand when a free convolution of two measures is absolutely continuous with respect to the Lebesgue measure. It is shown that if the support of a probability measure is contained in the positive half line, and is bounded away from zero, then its free multiplicative convolution with the semicircle law is absolutely continuous. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is needed.

This talk is based on joint works with Rajat Subhra Hazra and Deepayan Sarkar. 2015 — 16 Eigenvalue attraction and disordered Hamiltonian density of states Ramis Movassagh (Massachusetts Institute of Technology) Monday, 9th of May 2016 at in Malott 406 Probability 1. Eigenvalue Attraction – Much work has been devoted to the understanding of the motion of eigenvalues in response to randomness. The folklore of random matrix analysis, especially in the case of Hermitian matrices, suggests that the eigenvalues of a perturbed matrix repel. We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. We apply the results to the Hatano-Nelson model, random perturbations of a fixed matrix, real stochastic processes with zero-mean and independent intervals and discuss open problems.

Reference :

Journal of Statistical Physics, Feb. 2016, Volume 162, Issue 3, pp 615-643
http://link.springer.com/article/10.1007/s10955-015-1424-5

2. Disordered Hamiltonian Density of States– The method of "Isotropic Entanglement" (IE), inspired by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue distribution of quantum many-body systems with generic interactions. At the heart is a "Slider", which interpolates between two extrema by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them isotropically. Isotropic means that the eigenvectors are in generic positions. We prove that the interpolation is universal. We then show that free probability theory also captures the density of states of the Anderson model with arbitrary disorder and with high accuracy. Theory will be illustrated by numerical experiments.

[Joint work with Alan Edelman]

References:

Phys. Rev. Lett. 107, 097205 (2011)
Phys. Rev. Lett. 109, 036403 (2012) Stable-like processes with indices greater than two Mathav Murugan (University of British Columbia) Monday, 2nd of May 2016 at in Malott 406 Probability In Euclidean space, symmetric stable process with index $\alpha \in (0,2)$ is obtained by a time change of Brownian motion using the subordinator of index $\alpha/2$. By a similar subordination, there exist stable-like random walks with indices greater than two on various fractals and fractal-like graphs. However, existing methods to prove transition probability estimates fail in this setting.

Davies' method is a technique introduced by E. B. Davies to obtain heat kernel upper bounds for uniformly elliptic operators in Euclidean space. This method was extended to general symmetric Markov semigroups by Carlen, Kusuoka and Stroock. In this talk, I will introduce some of the ideas involved in extending Davies method to obtain heat kernel bounds for stable-like processes with indices greater than two.

This talk is based on joint works with Laurent Saloff-Coste. Stein's method and distributional fixed point biasing with application to preferential attachment random graphs Erol Peköz (Boston University) Monday, 25th of April 2016 at in Malott 406 Probability Stein's method is used to get error bounds for distributional limit theorems in difficult settings with dependence. One variant of the method uses a distributional fixed point equation for the limit to create biased random variables that couple closely and give Kolmogorov error bounds. The preferential attachment random graph model is used in network science to model growth of networks where heavily connected nodes attract more future connections; this is a setting where there has been recent progress using this variant of Stein's method. I will survey some of this progress and give some new results in the multivariate approximation setting. Kolmogorov extension, martingale convergence, and compositionality of processes Dexter Kozen (Cornell University) Monday, 18th of April 2016 at in Malott 406 Probability We show that the Kolmogorov extension theorem and the Doob martingale convergence theorem are two aspects of a common generalization, namely a colimit-like construction in a category of Radon spaces and reversible Markov kernels. The construction provides a compositional denotational semantics for standard iteration operators in probabilistic programming languages, e.g. Kleene star or while loops, as a limit of finite approximants, even in the absence of a natural partial order. Hybrid percolation, spatial epidemics, and their measure-valued scaling limits Steve Lalley (University of Chicago) Monday, 11th of April 2016 at in Malott 406 Probability Hybrid percolation is a random graph on the vertex set $V= \mathbb{Z}^{d}\times [N]$ in which vertices $(x,k)$ and $(y,l)$ are connected with probability $p_{N}$ if $x$ and $y$ are neighboring vertices of the lattice $\mathbb{Z}^{d}$. The percolation graph is the skeleton of a spatial epidemic that evolves as follows: (a) infected individuals, represented by vertices of the graph, remain infected for one unit of time, after which they recover and acquire permanent immunity from further infection; and (b) infected individuals infect susceptible individuals (vertices at neighboring sites of $\mathbb{Z}^{d}$) with probability $p_{N}$. We shall describe results concerning the scaling limits of both the percolation graph and the associated epidemic process as $N$ becomes large in the critical regime $p_{N} = 1/(2dN)$. Expansion of random graphs Doron Puder (Institute for Advanced Study) Monday, 4th of April 2016 at in Malott 406 Probability We present an approach to showing that random graphs are nearly optimal expanders. This approach is based on deep results from combinatorial group theory. It applies to both regular and irregular random graphs.

In the talk, I will focus on regular graphs. Let $G$ be a random $d$-regular graph on $n$ vertices. It was conjectured by Alon (86') and proved by Friedman (08') in a ~100 page-long booklet that the highest non-trivial eigenvalue of $G$ is a.a.s. arbitrarily close to $2\sqrt{d-1}$. Our proof is substantially simpler and nearly recovers Friedman’s result. I will present the main ideas that go into the proof, some of them of independent interest. Parabolic Harnack inequalities on Dirichlet spaces Janna Lierl (University of Illinois at Urbana-Champaign) Monday, 21st of March 2016 at in Malott 406 Probability I will present recent results on applying the parabolic Moser iteration method in the setting of (fractal-type) metric measure Dirichlet spaces. Under appropriate geometric conditions, we obtain that non-negative local weak solutions to the heat equation are locally bounded, Hölder continuous, and satisfy a strong parabolic Harnack inequality. The parabolic Harnack inequality is known to characterize heat kernel estimates of (sub-)Gaussian type. An application of this equivalence, together with the Doob's transform technique, allow to obtain sharp bounds for the Dirichlet heat kernel on inner uniform domains. Random walk with site-based feedback Nick Travers (Indiana University, Bloomington) Monday, 14th of March 2016 at in Malott 406 Probability We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$.

We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either $R=1$ or $L=1$ and when $R=L=2$. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed. ​Our model is related to the excited random walk model introduced by Benjamini and Wilson, and some extensions thereof. Many of the same techniques from the study of excited random walks are used in our analysis as well. Joint work with Ross Pinsky. Boundary Harnack principle and Martin boundary at infinity for Feller processes Zoran Vondracek (University of Illinois at Urbana-Champaign) Monday, 7th of March 2016 at in Malott 406 Probability A boundary Harnack principle (BHP) was recently proved for Feller processes in metric measure spaces by Bogdan, Kumagai and Kwasnicki. In this talk I will first show how their method can be modified to obtain a BHP at infinity – a result which roughly says that two non-negative function which are harmonic in an unbounded set decay at the same rate at infinity. With BHP at hand, one can identify the Martin boundary of an unbounded set at infinity with a single Martin boundary point and show that, in case infinity is accessible, this point is minimal. I will also present analogous result for a finite Martin boundary point. The local character of these results implies that minimal thinness of a set at a minimal Martin boundary point is also a local property of that set near the boundary point. Joint work with P.Kim and R.Song. Spectral thresholds in the bipartite stochastic block model Laura Florescu (Courant Institute at New York University) Monday, 29th of February 2016 at in Malott 406 Probability We consider a bipartite stochastic block model on vertex sets $V_1$ and $V_2$ of size $n_1$ and $n_2$ respectively, with planted partitions in each, and ask at what densities can spectral algorithms recover the partition of the smaller vertex set. The model was previously used to give a unified algorithm for random planted hypergraph partitioning and planted random k-SAT.

When $n_2 \gg n_1$, multiple thresholds emerge, and we propose a simple spectral algorithm, Diagonal Deletion SVD, which recovers the partition at a lower density than the usual SVD. We also locate a sharp threshold for detection of the partition, in the sense of the results of Mossel, Neeman, Sly and Massoulié for the stochastic block model. This gives the best known bounds for efficient recovery densities in planted k-SAT and hypergraph partitioning as well as showing a barrier to further improvement via the reduction to the bipartite block model. Joint work with Will Perkins. Synchronization of finite-state pulse-coupled oscillators on various graphs Hanbaek Lyu (The Ohio State University) Monday, 22nd of February 2016 at in Malott 406 Probability We propose a novel generalized cellular automaton (GCA) model for discrete-time pulse-coupled oscillators and study the emergence of synchrony. Our discrete model brings the study of coupled oscillators into the realm of combinatorics and probability. Given a finite simple graph and an integer $n\ge 3$, we place identical $n$-state oscillators on vertices with the following weak coupling along the edges: each oscillator inhibits its phase update if it has at least one neighboring oscillator at a particular "blinking" state and if its state is ahead of this blinking state. We obtain conditions on initial configurations and on network topologies for which states of all vertices eventually synchronize. For finite graphs, we show that our GCA model synchronizes arbitrary initial configurations on paths, trees, and, with random perturbation, any connected graph. In particular, we obtain the following local-global principle for tree networks: for $n\in \{3,4,5,6\}$, any $n$-periodic network on a tree synchronizes arbitrary initial configuration if and only if the maximum degree of the tree is less than $n$.

For infinite graphs, we study our discrete dynamics on the integer lattice $\mathbb{Z}^{d}$ using probabilistic methods. For $d=1$, our model is related to annihilating particle systems and random walks, and by this connection, we show that starting with uniform product measure, the probability of not having synchrony on a finite fixed interval at time $t$ decays to zero in the order of $O(t^{-1/2+o(1)})$. For $d\geq 2$ and for $n\neq 4$, our model shows spiraling autowave behavior, and hence non-synchrony, in resemblance of the famous Belousov–Zhabotinsky reaction. While this behavior is common to similar models for excitable media, the four-color model is critical and characteristic behavior: we seem to have clustering on $\mathbb{Z}^{d}$ through a complex wave process, regardless of dimension. This may be thought of as a four-state Ising spin system with deterministic dynamics, with convergence toward global ferromagnetism.

If time allows, we also discuss an asynchronous and continuous-state generalization of our GCA model. Despite being continuous model, we can prove similar synchronization theorems for trees. As an application, we combine this model with a distributed spanning tree algorithm and obtain a distributed self-stabilizing algorithm for clock synchronization in a system of processors with identical local clocks on any connected graph with $v$ nodes and diameter $d$, which runs in $O(d+\sqrt{v}\log^{*}v)$ time. Mixing on unipotent groups Bob Hough (Institute for Advanced Study) Monday, 8th of February 2016 at in Malott 406 Probability I will discuss a new local limit theorem on the Heisenberg group which applies to arbitrary measures of compact support and, under a mild condition on the characteristic function, obtains an optimal rate of convergence. The method extends to give mixing results for a natural class of random walks on the group $U_n(\mathbb{Z}/p\mathbb{Z})$ of $n \times n$ upper triangular matrices with entries from $\mathbb{Z}/p\mathbb{Z}$. Joint work with Persi Diaconis. Limit theorems for monotone subsequences in Mallows permutations Nayantara Bhatnagar (University of Delaware) Monday, 1st of February 2016 at in Malott 406 Probability The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is $2\sqrt{n}$. This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution.

We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $p$ in $S_n$ is proportional to $q^{\text{Inv}(p)}$ where $q$ is a real parameter and $\text{Inv}(p)$ is the number of inversions in $p$. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter $q$. In the regime that $q$ is constant, we make use of the regenerative structure of the permutation to prove a Gaussian CLT for the LIS.

This is based on joint work with Ron Peled and with Riddhi Basu. Combinatorics of the two-species ASEP and generalizations Olya Mandelshtam (University of California, Berkeley) Monday, 30th of November 2015 at in Malott 406 Probability The asymmetric exclusion process (ASEP) is an important and well-studied non-equilibrium model from statistical physics, in which particles hop left and right on a finite one-dimensional lattice with open boundaries. Typically the model has 3 hopping parameters (governing the rates at which particles enter and exit the lattice, and hop left and right). The two-species ASEP is a generalization of the ASEP in which there are two types of particles, one "heavy" and one "light." Much past work was devoted to finding combinatorial formulas for the steady state probabilities of the original ASEP. In this talk I will present our recent work on the combinatorics of the two-species ASEP. Together with Viennot, we introduced "rhombic alternative tableaux," which are fillings of certain rhombic tilings, and used them to provide combinatorial formulas for the two-species ASEP. Furthermore, in recent joint work with Corteel and Williams, we introduced "rhombic staircase tableaux," and provided combinatorial formulas for the more general 5-parameter two-species ASEP, which has an interesting connection to Koornwinder-Macdonald polynomials. The talk will be accessible to graduate students. Analyticity of drift and entropy of random walks on regular languages Lorenz Gilch (Technische Universität Graz) Tuesday, 24th of November 2015 at in Malott 205 Probability In this talk we will consider random walks with bounded range on regular languages over a finite alphabet. The random walks can be described by a finite set of parameters. The main goal of this talk is to show that drift and asymptotic entropy of transient random walks vary real-analytically in terms of probability measures of constant support. For this purpose, I will give a brief introduction to random walks on regular languages and I will give formulas for the drift and entropy, from which one can deduce analyticity. The main idea is to cut the random walk into pieces such that this sequence of pieces form a hidden Markov chain, from which we get the analytic behaviour of the entropy. Finally, I will consider positive recurrent random walks and I will show how to compute the entropy in that case explicitly. Current large deviations in the boundary-driven symmetric simple exclusion process on the Sierpinski gasket Joe Chen (University of Connecticut) Monday, 23rd of November 2015 at in Malott 406 Probability We study the symmetric simple exclusion process on the Sierpinski gasket ($SG$) driven by the action of particle reservoirs attached to boundary vertices of $SG$.

We establish three hydrodynamic limit theorems for the empirical current: the law of large numbers, the large deviations principle, and the large deviations principle for the mean current on a long-time interval. On $\mathbb{Z}^d$ these results were established assuming translational invariance and Gaussian space-time diffusive estimates. But on $SG$ neither assumption is valid, and it is unclear how to make sense of the resulting reaction-diffusion PDE.

In this work we overcome all the aforementioned obstacles. First, we establish the "moving particle lemma" on weighted graphs, using the "octopus inequality" of Caputo, Liggett, and Richthammer in their seminal proof of Aldous' spectral gap conjecture. This enables us to prove a local version of the two-blocks estimate on $SG$, thereby answering a question posed by Jara. Second, we actively use the theory of differential $1$-forms on $SG$ developed by Teplyaev and collaborators, which allows us to characterize the speed of convergence of discrete $1$-forms on $SG$, and prove uniqueness of solutions to the resulting reaction-diffusion PDE.

This is joint work with Alexander Teplyaev. Inequalities for critical exponents in $d$-dimensional sandpiles Jack Hanson (Georgia Institute of Technology) Monday, 23rd of November 2015 at in Malott 406 Probability Consider the Abelian sandpile measure on $\mathbb{Z}^d$, $d\geq 2$, obtained as the $L\rightarrow\infty$ limit of the stationary distribution of the sandpile on $[-L, L]^d$. When adding a grain of sand at the origin, some region topples during stabilization. We prove bounds on the tail behavior of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. Braess’s paradox for the spectral gap in random graphs and delocalization of eigenvectors Tselil Schramm (University of California, Berkeley) Monday, 16th of November 2015 at in Malott 406 Probability Braess's paradox is a famous phenomenon from game theory, stating that in traffic networks, the addition of a new road can sometimes worsen congestion. I will describe a similar phenomenon in random graphs: for typical instances of Erdős–Rényi random graphs $G(n,p)$ with constant edge density $p\in(0,1)$, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. I will show that this problem reduces to a question about the delocalization of eigenvectors of normalized Laplacians of $G(n,p)$, and then describe our new eigenvector delocalization results.

Based on joint work with Ronen Eldan and Miki Racz Phase transition for the threshold contact process, an "annealed approximation" of heterogeneous random Boolean networks Shirshendu Chatterjee (The City University of New York) Monday, 9th of November 2015 at in Malott 406 Probability We consider a model for heterogeneous gene regulatory networks that is an "annealed approximation" of Kauffmann's (1969) original random Boolean networks. In this model, genes are represented by the nodes of a random directed graph $G_n$ on $n$ vertices with specified degree distribution, and the interactions among the genes are approximated by an appropriate threshold contact process (in which a vertex with at least one occupied in-neighbor at time $t$ will be occupied at time $t+1$ with probability $q$, and vacant otherwise) on $G_n$. We characterize the order-chaos phase transition curve for the threshold-contact process on $G_n$ segregating the chaotic and ordered random Boolean networks. Finitely dependent graph homomorphisms Avi Levy (University of Washington) Monday, 2nd of November 2015 at in Malott 406 Probability When a child randomly paints a coloring book, adjacent regions receive distinct colors whereas distant regions remain independent. It took mathematicians until 2014 to replicate this effect, when Holroyd and Liggett discovered the first stationary $k$-dependent $q$-colorings. In this talk, I will discuss an extension of Holroyd and Liggett's construction which associates a canonical insertion procedure to every finite graph. The known colorings turn out to be diamonds in the rough: apart from multipartite analogues, they are the only $k$-dependent processes which arise from finite graphs in this manner. Time permitting, I will present extensions of these results to weighted graphs and shifts of finite type. Joint work with Alexander Holroyd. Longest increasing path within the critical strip Partha Dey (University of Illinois at Urbana-Champaign) Monday, 26th of October 2015 at in Malott 406 Probability Consider a Poisson Point Process of intensity one in the two-dimensional square $[0,n]^2$. In Baik-Deift-Johansson (1999), it was shown that the length $L_n$ of a longest increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later Johansson (2000) showed that all maximal paths lie within the strip of width $n^{2/3+\epsilon}$ around the diagonal with probability tending to $1$ as $n\to \infty$. We consider the length $L_n^{(\gamma)}$ of maximal increasing paths restricted to lie within a strip of width $n^{\gamma}, \gamma<2/3$ around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution. We also obtain tight bounds on the expectation and variance of $L_n^{(\gamma)}$. Joint work with Matthew Joseph and Ron Peled. Non-uniqueness in SPDEs Yu-Ting Chen (Harvard University) Monday, 19th of October 2015 at in Malott 406 Probability One difficult problem for uniqueness in SPDEs, open for more than two decades, is to determine pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion for nonnegative solutions. A recent work by Mueller, Mytnik and Perkins disproves, however, the stronger conjecture that the square-root diffusion coefficient of the SPDE is a robust mechanism to keep solutions of SPDEs unique in general, among other things. The counterexample studies an SPDE with the same coefficients as the SPDE of super-Brownian motion and signed solutions are allowed.

In this talk, I will discuss a result which continues the investigation of the SPDE of super-Brownian motion by Mueller et al. Only nonnegative solutions are considered, and now we introduce perturbation to the SPDE, which is realized by additional immigration mechanisms. I will first review the SPDE of super-Brownian motion and related aspects. I will then introduce SPDEs for super-Brownian motions with immigration and discuss a non-uniqueness result and its proof. The dimer model on the hexagonal lattice Sevak Mkrtchyan (University of Rochester) Monday, 5th of October 2015 at in Malott 406 Probability The dimer model is the study of random perfect matchings on graphs, and has a long history in statistical mechanics. On the hexagonal lattice it is equivalent to tilings of the plane by lozenges and to 3D stepped surfaces called skew plane partitions - 3 dimensional analogues of Young diagrams with a partition removed from the corner. We will discuss the scaling limit of the model under a certain family of measures called "volume"-measures, the limit-shape phenomenon in this model (a form of the law of large numbers), the effects of varying the boundary conditions on the limit shape, and the nature of local fluctuations in various regions of the limit shape. We will also discuss the behaviour of the system when the measure is modified in certain ways. The Parisi variational problem: a new approach Aukosh Jagannath (Courant Institute of Mathematical Sciences) Monday, 28th of September 2015 at in Malott 406 Probability The Parisi Variational Problem is a challenging non-local, strictly convex variational problem over the space of probability measures whose analysis is of great interest to the study of mean field spin glasses. In this talk, I will present a new, conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We will begin with a new characterization of the minimizers of this problem whose origin lies in the first order optimality conditions for this functional. As a demonstration of the power of this new approach, we will study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We will generalize their conjecture to all mixed p-spin glasses and prove that their condition is correct in the entire temperature-external field plane except for a compact set whose phase is unknown at this level of generality. A key element of this analysis is a new class of estimates regarding gaussian integrals in the large noise limit called "Dispersive Estimates of Gaussians". This is joint work with Ian Tobasco (NYU). Fourientations and the Tutte polynomial Sam Hopkins (Massachusetts Institute of Technology) Monday, 21st of September 2015 at in Malott 406 Probability A fourientation of a graph is a choice for each edge whether to orient that edge one way or another, to leave it unoriented, or to bidirect it. In joint work with Spencer Backman we produce a list of 64 classes of fourientations defined by cuts and cycles that are enumerated by generalized Tutte polynomial evaluations. Our result unifies and extends results due to many authors starting with Stanley's celebrated theorem that the number of acyclic orientations of a graph is $T(2,0)$. I will explain how I entered into this project via the study of the bigraphical hyperplane arrangement. Time permitting, I may also discuss connections to graphic Lawrence ideals, monomizations of power ideals and zonotopal algebras, or combinatorial Riemann-Roch theory. At the end I will also mention future work involving "fourientation activities". Fluctuation results for Hastings-Levitov planar growth Vittoria Silvestri (University of Cambridge) Monday, 14th of September 2015 at in Malott 406 Probability I will discuss one instance of the so called Hastings-Levitov planar aggregation model, consisting of growing random clusters on the complex plane, built by iterated composition of random conformal maps. In 2012 Norris and Turner proved that in the case of i.i.d. maps the limiting shape of these clusters is a disc. In this talk I will show that the fluctuations around this asymptotic behaviour are given by a random holomorphic Gaussian field $F$ on $\{\left|z\right| > 1\}$, of which I will provide an explicit construction. The boundary values of $F$ perform a Gaussian Markov process on the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of the whole plane Gaussian Free Field to the unit circle. 2014 — 15 Nucleation scaling in jigsaw percolation David Sivakoff (The Ohio State University) Monday, 4th of May 2015 at in Malott 406 Probability Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdős-Rényi graph with edge probability $p$, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition $\{V\}$. In some generality, for puzzle graphs with $N$ vertices of degrees about $D$, this probability is close to $1$ or $0$ depending on whether $pD\log N$ is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions. Poisson boundaries of random walks on Baumslag-Solitar groups Ecaterina Sava-Huss (Cornell University) Monday, 27th of April 2015 at in Malott 406 Probability I will start with a gentle introduction on Poisson boundaries for random walks on groups, and give two basic examples. Then I will consider a class of groups on two generators and one relation, which play an important role in geometric group theory. The groups under consideration are called Baumslag-Solitar groups. I will consider random walks driven by a probability measure with finite first moment on such groups, and study the behaviour at infinity on them: convergence to the boundary and the Poisson boundary. Statistical matching theory Péter Csikvári (Massachusetts Institute of Technology) Monday, 20th of April 2015 at in Malott 406 Probability In this talk we will survey some recent development on statistical properties of matchings of very large and infinite graphs. The main goal of the talk is to describe a few applications of a new concept called matching measure. These applications include new results on the number of (perfect) matchings in large girth graphs as well as simple new proofs of certain statistical physical theorems. In particular, we will sketch the proof of Friedland's Lower Matching Conjecture, and a new proof of Schrijver's and Gurvits's theorems. This talk is based on joint papers with various subsets of Miklos Abert, Peter E. Frenkel, Tamas Hubai and Gabor Kun. Every concept will be defined. The causes of metastability and their effects on transition times Katherine Newhall (University of North Carolina at Chapel Hill) Monday, 13th of April 2015 at in Malott 406 Probability Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. Rotor router walks on Galton-Watson trees Wilfried Huss (Cornell University) Monday, 6th of April 2015 at in Malott 406 Probability A rotor router walk is a deterministic walk on a graph, where the exits from each vertex follow a fixed cyclic order. We prove a necessary and sufficient condition for recurrence of rotor router walks on supercritical Galton-Watson trees, in the case where the first exit of each vertex is choosen at random.

Joint work with Sebastian Müller and Ecaterina Sava-Huss. Characterization of cutoff for reversible Markov chains Jonathan Hermon (University of California, Berkeley) Monday, 23rd of March 2015 at in Malott 406 Probability A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least $\alpha$, for some $0<\alpha<1$.

We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to infinity.

Joint work with Riddhipratim Basu and Yuval Peres. Random walks and isoperimetric profiles of groups Tianyi Zheng (Stanford University) Monday, 16th of March 2015 at in Malott 406 Probability $L^2$-isoperimetric profile was introduced by Grigor’yan in the form of Faber–Krahn inequalities on manifolds. The theory was further developed in his joint work with Coulhon and Pittet to establish the close relation between $L^2$-isoperimetric profile and decay of heat semigroup on general Riemannian manifolds and infinite graphs. By studying the isoperimetric profiles, we show that random walks exhibit rich exotic behavior on Neumann-Segal type branch groups. I will also discuss a general result that bounds the growth of entropy of random walks in terms of the return probability. Hitting questions and multiple points for stochastic PDE (SPDE) in the critical case Carl Mueller (University of Rochester) Monday, 9th of March 2015 at in Malott 406 Probability Hitting questions play a central role in the theory of Markov processes. For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. There has also been a huge amount of work on the related question of when a process has multiple points.

For SPDE, much less is known, but there have been a growing number of papers on the topic in recent years. Potential theory provides an answer in principle. But unfortunately, solutions to SPDE are infinite dimensional processes, and the potential theory is intractible. As usual, the critical case is the most difficult.

We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case. Hypocoercive diffusions Fabrice Baudoin (Purdue University) Monday, 2nd of March 2015 at in Malott 406 Probability In this talk, we will present a new method to study the convergence to equilibrium of hypocoercive diffusions. The method is based on local computations and parallels the Bakry-Emery approach to hypercontractivity. Phenomena of critical regimes in invariance principles for operator-scaling Gaussian random fields Yizao Wang (University of Cincinnati) Monday, 9th of February 2015 at in Malott 406 Probability Hammond and Sheffield (2013) recently introduced a model of correlated random walk of which the partial sums scale to a fractional Brownian motion with long-range dependence. In this talk, we consider a natural generalization of this model to higher dimensions. We define a $\mathbb{Z}^{d}$-indexed random field with dependence determined by an underlying random graph, and we study the scaling limit of its partial sums over increasing rectangles. An interesting phenomenon occurs: when the rectangles increase at different rates, different limiting fields may arise. In particular, there is a critical regime where the limiting field is operator-scaling, while this is not the case in other regimes.

Joint work with Hermine Biermé and Olivier Durieu. Stein's method for diffusion approximations of many-server queues Anton Braverman (Cornell University) Monday, 2nd of February 2015 at in Malott 406 Probability Diffusion approximations for many-server queues have been studied extensively since the pioneering work of Halfin and Whitt (1981). We focus on many-server queues that have phase-type service time distributions and exponential customer patience distributions. We establish rate of convergence of the steady-steady distribution of a many-server queue in the Halfin-Whitt regime. Our proof technique connects naturally with Stein's method for establishing normal approximations. This connection allows us to extend the recent work of Gurvich (2014).

This is the joint work with Jim Dai at Cornell University.