Workshop

We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in the study of some random matrix ensembles and so-called beta-ensembles. We will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes will be described. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. Based on joint works with Etienne Sandier, Nicolas Rougerie, Mircea Petrache, Thomas Leblé, Florent Bekerman and Scott Armstrong. on the statistical mechanics of systems of points with logarithmicor Coulomb interactions. After listing some motivations, we describe the “electric approach"which allows to get concentration results, Central Limit Theorems for fluctuations, and aLarge Deviations Principle expressed in terms of the microscopic state of the system.

When it is infeasible to solve a large linear system directly by inversion, light and scalable randomized iterative methods can be used instead, such as, Randomized Kaczmarz (RK) algorithm, or Stochastic Gradient Descent (SGD). I will discuss some cases when the questions from the non-asymptotic random matrix theory are inherent for proving convergence guarantees for these methods. This includes QuantileRK and QuantileSGD methods proposed for the systems that might contain large, sparse, potentially adversarial corruptions. Unlike the classical extensions of RK/SGD to noisy (inconsistent) systems, the new methods learn to avoid corruptions rather than tolerate the small noise, and result in exact convergence when up to one half of the equations can be arbitrarily corrupted. The second case is connected to the sketching techniques that considerably speed up the iterative solvers. Based on the joint work with Deanna Needell, Jamie Haddock and Will Swartworth.

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr (2018). In addition, we obtain asymptotics at infinity for the Painlevé transcendents and large gap asymptotics for the corresponding point processes. This is a joint work with Mattia Cafasso (Univ. Angers) and Tom Claeys (UC Louvain).

Computing the eigenvalues and eigenvectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. In this talk, we will focus on the spiked covariance matrix model, a popular and sophisticated model proposed by Johnstone. We will present some recent results on the limiting behavior of the extreme eigenvalues and eigenvectors of the spiked covariance matrices in the supercritical case. This talk is based on joint work with Zhigang Bao, Xiucai Ding, and Jingming Wang.

The directed landscape is the conejctured scaling limit of most natural random metrics on the plane. This random geometry seems to have its own axioms and postulates; some are like Euclid's and some entirely different. I will present three simple postulates and explain how to prove them.

The statistical behavior of random tilings of large domains has been an intense topic of mathematical research for decades, partly since they highlight a central phenomenon in physics: local behaviors of highly correlated systems can be very sensitive to boundary conditions. Indeed, a salient feature of random tiling models is that the local densities of tiles can differ considerably in different regions of the domain, depending on the shape of the domain. Thus, a question of interest is how the shape of the domain affects the statistics of a random tiling. A general prediction in this context is that, while the family of possible domains is infinite-dimensional, the limiting statistics should only depend on a few parameters. In this talk, we will explain this universality phenomenon in the context of random lozenge tilings (equivalently, the dimer model on the hexagonal lattice). Various possible limiting local statistics can arise, depending on whether one probes the bulk of the domain; the edge of a facet; or the boundary of the domain. We will describe recent works that verify the universality of these statistics in each of these regimes.

These results are based on join works with Vadim Gorin and Jiaoyang Huang.

Consider the KPZ equation on an spatial interval [0,1] with mixed Neumann boundary conditions at 0 and 1. For each given pairs of boundary parameters (u,v), there should exist a unique stationary measure for the height profile differences (i.e., for the derivative of the KPZ equation). In this talk I will describe recent work in which we show that for each pair (u,v) satisfying u+v>0, certain exponentially reweighted Brownian paths measures are stationary measures for the corresponding open KPZ equation. Along the way, we will also encounter the open ASEP, as well as Askey-Wilson processes and q-function asymptotics. This is mainly based on my recent work with Alisa Knizel, though also relies on earlier work with Hao Shen as well as earlier work of Wlodzimierz Bryc, Jacek Wesolowski and Yizao Wang. I will also touch on some recent related work of Wlodzimierz Bryc, Alexey Kuznetsov, Jacek Wesolowski and Yizao Wang; as well as work of Guillaume Barraquand and Pierre Le Doussal.

Many complex statistical mechanical models have intricate energy landscapes. The ground state, or lowest energy state, lies at the base of the deepest valley. In examples such as spin glasses and Gaussian polymers, there are many valleys; the abundance of near-ground states (at the base of valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the model's disorder is slightly perturbed. Indeed, a monograph of Sourav Chatterjee from 2014 establishes that, for a class of models of Gaussian disorder, this abundance of competing minimizers is accompanied both by a rapid outset of chaos under perturbation of the system by noise, and by the effect of  super-concentration, in which model statistics have lower variance than in classical scenarios, for which a central limit theorem may apply. In this talk, a recent investigation, jointly undertaken with Shirshendu Ganguly, of a natural dynamics for a model of planar last passage percolation will be discussed. Robust probabilistic and geometric technique permits a very quantified analysis of the presence of close rivals in energy to the ground state for the static version of the model; consequently, the order of the scale that heralds the transition from stability to chaos for the dynamical model is identified. The tools that drive the investigation include harmonic analytic technique present in Chatterjee's work, and the use of Brownian Gibbs resampling analysis for random ensembles of curves naturally associated to last passage percolation via the Robinson-Schensted-Knuth correspondence.