Todd Kemp (UCSD): Matrix Random Walks, Brownian Motion, and the Lima Bean Law

Abstract: It has been a long-standing open problem to understand the high-dimensional limit behavior of eigenvalues of Brownian motion on the group of invertible matrices. Such random matrices are almost surely non-normal, which makes the task quite difficult due to the unstable properties of pseudo-spectrum. Over the last decade, we have identified what the limit should be, but the tools to prove convergence evaded us.

In two recent papers with overlapping authors, we have fully resolved this problem. Moreover, the eigenvalues of a broad family of matrix random walk approximations to Brownian motion have also been fully analyzed, and produced an unexpectedly powerful new scaling limit law: almost regardless of covariance, if the random walk steps are unitarily bi-invariant, the limit eigenvalue distribution always matches the Brownian motion (and in fact exhibits superconvergence). When the process starts at the identity matrix, the spectrum looks like a lima bean.

This is joint work with Tanya Brailovskaya, Nick Cook, Bruce Driver, Brian Hall, Ching Wei Ho, Yuriy Nemish, Vaki Nikitopoulos, and Felix Parraud.

Haotian Gu (UCLA): Maximum of Poissonian Log-correlated Fields

Abstract: Extreme values of logarithmically correlated fields (LCFs) have received lots of interests due to connections with Gaussian multiplicative chaos, random matrices, branching random walks, reaction-diffusion PDEs, and L-functions in analytic number theory. The sharpest results are for Gaussian or nearly-Gaussian fields. On the other hand, characteristic polynomials of sparse random matrices give rise to LCFs with Poissonian tails. In an earlier work on permutation matrices, Cook and Zeitouni obtained the leading order of the maximum. I will discuss new refined results on the maximum for a related class of random function series with Poissonian tails. We find the sub-leading order behavior is significantly different from the ubiquitous "Bramson correction" term for Gaussian LCFs, and can be modeled by a branching random walk in a random time-inhomogeneous environment. Based on upcoming joint work with Nicholas Cook (Duke).

Yujin Kim (Caltech): The low temperature SOS model above a wall and 1:2:3 scaling

Abstract: Random surfaces have played a central role in probability and mathematical physics for decades. Physically, they often arise as models of interfaces: boundaries between distinct regions of space. The Solid-on-Solid (SOS) model is a canonical model for interfaces separating stable (equilibrium) coexisting phases in three dimensions, such as the boundary of a solid that has crystallized in a liquid solution. In this talk, we present the fascinating geometry of the SOS model at "low temperature", conditioned to be non-negative ("above a wall": think of a crystal growing on a lab slide, so that the crystal cannot grow downwards). In this setting, the SOS model resembles a wedding cake, being comprised of a sequence of shrinking, stacked layers whose boundaries form a collection of nested loops. Our work sheds light on the fluctuations of these loops away from their Wulff shape scaling limits, and in particular suggests a scaling limit for these fluctuations. Joint works with Patrizio Caddeo, Milind Hegde, Eyal Lubetzky, and Christian Serio.

Pablo Lopez Rivera (UCLA): Stochastic processes, transport of mass, and functional inequalities

Abstract: Functional inequalities have proven to be a ubiquitous tool in mathematics, especially in probability theory. For example, they are closely related to the concentration of measure phenomenon, and they help quantify the rate at which ergodic Markov processes converge to equilibrium. Prominent examples of those inequalities include the families of logarithmic Sobolev, Poincare, and transport-entropy inequalities. In the first part of the talk, I will provide an introduction to this topic, highlighting the classical examples, results, and applications.

In the second part of the talk, I will address the following question: If we perturb a known measure that satisfies a functional inequality, is the original functional inequality still valid for the perturbed measure? In the Gaussian setting, the theory of optimal transport provides an affirmative answer, thanks to Caffarelli's contraction theorem. In this talk, I will show how we can generalize Caffarelli's theorem to both the smooth and discrete settings, with the help from some stochastic processes that will allow us to construct (deterministic) transport maps that will permit the transfer of functional inequalities.

Lingfu Zhang (Caltech): Sharp phase transition in the repeated averaging process

Abstract: Consider a connected finite graph with a real number assigned to each vertex. At each step, an edge (u,v) is chosen uniformly at random, and the numbers at u and v are replaced by their average. This dynamics, known as the repeated averaging process, arises in many contexts, such as thermal equilibration, opinion dynamics, wealth exchange, and quantum circuits. All numbers eventually converge to the global average, and we study the speed of convergence in the L¹ distance (which corresponds, for example, to the Gini index in wealth distributions). On random d-regular graphs, we show that this decay undergoes a sharp phase transition, with the L¹ distance dropping abruptly to zero according to a Gaussian profile. Our techniques are robust, and we expect them to extend to more general dynamics on expander graphs. This is joint upcoming work with Dong Yao.

Zhengye Zhou (USC): Uniform convergence of Pfaffian Point Process

Abstract: Recent years have seen significant progress in understanding fluctuations in half-space models within the KPZ universality class. In this talk I will discuss a Pfaffian Schur process, which is a measure on a sequence of partitions, which was introduced by Borodin and Rains as a Pfaffian analog of the determinantal Schur processes introduced by Okounkov and Reshetikhin. The model we investigate arises in a half-quadrant last passage percolation model, which has i.i.d. geometric weights with parameter α∈(0,1) off of the main diagonal and with parameter c∈ (0, α^-1) on the main diagonal. We show that when c is subcritical the line ensembles formed by the parts of our random partitions converge uniformly over compact sets to the Airy line ensemble; when c is critically scaled with the size of the system the ensemble instead converges to the Airy wanderer line ensemble with a single parameter; and when c is supercritical we will see both Gaussian and Airy line ensemble in the limit