ICTS

We study the evolution of a random string in  Poisson trap environment and compute the exponents of the limiting survival probability.  It is well known that when the obstacles are hard then the problem is directly related to the sausage volume of the string.  We shall briefly review the literature on particles moving in a trap environment, explain the key strategies that the particle may employ to survive and then discuss the same for the random string. This is joint work with Carl Mueller and Mathew Joseph

We consider weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet, where the bulk of the particles accumulate in the many-particle limit. For Coulomb gases at determinantal inverse temperature, and zeros of random polynomials, we observe that the limiting outlier process only depends on the shape of the uncharged region containing them (and the global net excess charge).

In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a demonstration of screening properties of the particle system.

Based on a joint work with R. Butez, D. García-Zelada, and A. Wennman.

The Brunn-Minkowski theory in convex geometry concerns, among other things, the study of volumes, mixed volumes, and surface area measures of convex bodies. We discuss the generalizations of these concepts to Borel measures with density in $\R^n$, with particular focus on the standard Gaussian measure. After reviewing the weighted surface area and mixed measures (which are the weighted versions of mixed volumes), we introduce the mixed measures of three convex bodies, and explore inequalities related to all these quantities. The talk is based on joint work with Dylan Langharst, Matthieu Fradelizi, and Artem Zvavitch.

The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analog of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behavior has remained out of reach. In this talk, based on joint work with Milind Hegde, we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach relies on a Gibbs resampling property of the KPZ equation as well as classical correlation inequalities, and yields a sharp understanding for a large class of initial data.

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko−Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas for the logarithmic energy of the mentioned models which can be of independent interest. Joint work with D. Chafaï et P. Youssef.

For various random matrix models of fairly general integral entries we discuss the probability that the cokernels are isomorphic to a given finite abelian group, or when they are cyclic. We will show that these statistics are asymptotically universal, given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra. 

Based on joint works with  R. V. Peski and  M. M. Wood.

I will discuss some examples and some general limit theorems for interacting models defined on spatial random graphs constructed on stationary Euclidean point processes in a finite window.  A prototypical example is co-operative sequential adsorption on the nearest neighbour graph built on the Ginibre point process. We establish the asymptotic normality, expectation, and variance asymptotics of statistics of the system evolution as the window size increases. I will try to indicate the general theory underlying these limit theorems. In particular, we will focus on the joint weak decay of correlations to be satisfied by the graph structure, point processes and initial states as well as sufficient conditions for the same. This is an on-going work with B. Blaszczyszyn (INRIA-ENS, Paris) and J. E. Yukich (Lehigh University, USA).

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. We prove that, for the walk started from a leaf vertex and stopped upon hitting the root, as the depth of the tree tends to infinity the maximal time spent at any leaf converges, under suitable scaling and centering, to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree. Joint work with Marek Biskup (UCLA).

We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

I will describe the extremal landscape for the characteristic polynomial and orthogonal polynomials associated with the CbetaE ensemble. In particular, it will follow that the maximum on the unit circle of the (absolute value) of the log-determinant converges in distribution to the sum of two independent variables, one of which is Gumbel.
Joint work with Elliot Paquette.

We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

It has been an active research topic to understand the spectral statistics of random matrices. An important and interesting question is investigating large deviation properties of spectral observables, such as the empirical spectral measure and extreme eigenvalues. In the absence of exact formulas such as in Gaussian ensembles, these questions are quite hard to analyze. In particular, for sparse random matrices, even investigating the typical behavior of eigenvalues is nontrivial. In this talk, we will survey recent progress in understanding spectral large deviations of such sparse random networks. Based on joint works with Shirshendu Ganguly and Ella Hiesmayr.

In this series of lectures, we will explore a medley of complex stochastic models and their applications to a range of problems from statistical physics to machine learning. These include continuous spin systems such as the Heisenberg and XY models, determinantal point processes and their generalizations, and the problem of learning under the action of latent group invariances. For continuous spin models, we will demonstrate a very general technique for obtaining Gaussian fluctuations at near-optimal rates, leveraging connections to the Lee-Yang theory of phase transitions and a quantitative generalization of the classical Marcinkiewicz Theorem. We will explore new kinds of determinantal processes arising from applications to the problems of stochastic gradient descent (SGD) and dimensionality reduction, and show how their stochastic geometry leads to connections with spiked models of random matrices and orthogonal polynomials. Finally, we will discuss the problem of learning under latent group actions, which is motivated by the well-known procedure of cryo-EM, and  leads to an interesting interplay of structural constraints (such as sparsity) with harmonic analysis, group representations and invariant theory.

The statics and dynamics of mean-field models of spin glasses have been studied in-depth by the physics community since the '70s. At the heart of this is the trade-off between the notions of replica symmetry breaking, shattering, and metastability. I will survey the current mathematical understanding of these ideas in the “simple” case of the spherical p-spin model. I will start by recalling how the landscape complexity can be used to understand of the “replica symmetry breaking” phase following the work of Auffinger–Ben Arous–Cerny and Subag. I'll then turn to our recent joint work with Ben Arous on the “replica symmetric” phase. Here we prove the existence of a shattering phase and show that metastable states exist up to an even higher temperature as predicted by Barrat–Burioni–Mezard. This latter work is based on a Thouless–Anderson–Palmer decomposition which builds on the ideas of Subag. I will end by presenting a series of open questions and conjectures surrounding sharp phase boundaries for shattering and metastability.
This talk will touch on joint work with: A. Auffinger (Northwestern), G. Ben Arous (Courant), R. Gheissari (Northwestern), and I. Tobasco (UIC)