Estimating the probability of rare events, namely large deviations principle, for the spectrum of Wigner matrices with compactly supported entries is a challenge. In this talk we will discuss how the asymptotics of Spherical integrals can be derived and used to study such problems. This is based on  recent joint works with Augeri, Belinschi, Huang and Husson.

The characteristic polynomial of the GUE is at the heart of a conjecture from Fyodorov and Simm, where it is crucial to understand the log-correlated structure of the field induced. As a first step in this direction, we obtain a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian $\beta$ Ensemble. Relying on the tridiagonal representation of such matrix models, we will explain how the second order recursion satisfied by the characteristic polynomial allows us to give a martingale representation of its logarithm, leading to an analysis of its fluctuations. This is a joint work with R. Butez and O. Zeitouni.

The Gaussian ensembles, originally introduced by Wigner may be generalized to an n-point ensemble called the beta-Hermite ensemble. As with the original ensembles we are interested in studying the local behavior of the eigenvalues. At the edges of the ensemble the rescaled eigenvalues converge to the Airy_beta process which for general beta is characterized as the eigenvalues of a certain random differential operator called the stochastic Airy operator (SAO). In this talk I will give a short introduction to the Stochastic Airy Operator and the proof of convergence of the eigenvalues, before introducing another interesting eigenvalue process. This process can be characterized as a limit of eigenvalues of minors of the tridiagonal matrix model associated to the beta-Hermite ensemble as well as the process formed by the eigenvalues of the SAO under a restriction of the domain. This is joint work with Angelica Gonzalez.

In the simple exclusion process on a finite segment, a particle is allowed to move to the right at rate $p$ and to the left at rate $q$, provided that the selected site is empty. In joint work with Nina Gantert and Domink Schmid, we study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on $p,q$ as well as on the entering and exiting rates, and show that the process exhibits pre-cutoff and in some special cases even cutoff.

Given a finite graph, the arboreal gas is the measure on spanning forests of this graph in which each edge of the forest is weighted by a parameter β>0. The main question is whether the arboreal gas percolates for a given β, i.e., if there is a component of the forest covering a positive fraction of the graph. We prove that (perhaps surprisingly) the arboreal gas does not percolate in two dimension, but that it does percolate in dimensions three and higher provided β>β_0. Our analysis also implies diffusive subleading corrections to the connection probabilities in the latter case. This problem has a lot of independent motivation from probability, combinatorics, and statistical physics, but what does it have to do with random matrices? Analogous behaviour is predicted for random Schroedinger operators or random band matrices. The arguably best physics predictions for this Anderson transition rely on the supersymmetric approach to random matrices. The question of percolation of the arboreal gas is also exactly related to that of spontaneous symmetry breaking of a (supersymmetric) nonlinear sigma model (the H02 model), but one that is vastly simpler to analyse. Indeed, while understanding spontaneously broken symmetries is in general a notorious problem, in the case of the H02 model, we can construct a low temperature renormalisation group flow in dimensions three and higher (and derive the percolation results from it). This is joint work with N. Crawford, T. Helmuth, and A. Swan.

Members of the Riesz family are statistical physics systems that include Coulomb and log-gases. Each one gives rise to a (Gibbsian) point process, and a basic question is to study the random number of points in large d-dimensional boxes. In some "solvable" cases, one finds interesting properties like hyperuniformity and (number)-rigidity as well as a tripartite "large deviations" estimate. "Universality" of such features with respect to the interaction, the temperature, the external field is mostly open. I will present several questions (and few results) around this theme.

We consider a two dimensional eigenvalue density which appears on the complex plane if we add a rank one imaginary perturbation to the Hermitian random matrix. The case of band matrices is of prime interest for us.

This talk will cover a natural generalization of moments of characteristic polynomials from the classical compact groups (unitary, symplectic, orthogonal). The results presented resolve a conjecture of Fyodorov and Keating. There are natural connections to log-correlated fields, integrable systems (asymptotics of Toeplitz and Hankel determinants, Painlevé), symmetric function theory, and number theory.

Fyodorov-Hiary-Keating proposed very precise asymptotics for the maximum of the Riemann zeta function in almost all intervals along the critical axis. After reviewing the origins of this conjecture through the random matrix analogy, I will explain a proof up to tightness, building on an underlying branching structure. This work with Louis-Pierre Arguin and Maksym Radziwill relies on a multiscale analysis and twisted moments of zeta.

Random graphs and hypergraphs have long been employed as network models for a bevy of machine learning problems, from clustering and community detection to coding theory, signal processing, and so on. The spectral properties of regular models, in particular things like the spectral gap, have often played an important role in such applications. In this wide-audience talk, I will mention some of the random matrix techniques used in the analysis of spectral gap in such regular structures, as well as some of the challenges and open problems.

The Erdös-Renyi graph G(N,d/N) is well known to undergo a dramatic change of behaviour around d of order log N. Below this threshold the graph develops inhomogeneities, which result in the emergence of a localized phase, in coexistence with a delocalized phase. I give an overview of the corresponding phase diagram, along with recent progress in establishing it rigorously. Joint work with Johannes Alt and Raphael Ducatez.