## Periodic Reporting for period 2 - LDRaM (Large Deviations in Random Matrix Theory)

Reporting period: 2022-03-01 to 2023-08-31

Large random matrices first appeared in the analysis of large arrays of data in the work of Wishart in the thirthy’s, in theoretical physics since the work of Wigner, and since then in many scientific domains, including for instance recently in statistical learning. They are also connected with many other domains such as random tilings or quantum chaos. In particular, a large deviations theory for large random matrices would have groundbreaking applications in many domains of mathematics, physics and statistics.

The classical results from large deviations theory do not apply to most relevant quantities in random matrix theory, such as the spectrum of a random matrix, because they are complicated functions of independent variables. During the last twenty years, important advances allowed to analyze the large deviations for very particular models of random matrices, but a full theory is still lacking. The goal of this project is to develop such a theory.

The first part of the project concerns the large deviations for the spectrum of large random matrices. The second considers precise large deviations for Coulomb Gases and more general mean field strongly interacting particles systems, including the so-called Sinh models from quantum physics, integrable quantum field theories or discrete models such as random tilings. The next objectives focus on multi-matrix models which arise in physics and combinatorics, including spherical integrals and unitary models, and reaches its climax with long standing open questions about Voiculescu’s entropy. The last objective concentrates on applications of large random matrices and large deviations in machine and statistical learning.

Precise large deviations for strongly interacting particle systems have been established in great generality. This includes models coming from physics such as Riesz gases for which the fluctuations of particles were studied for the first time rigorously. In a new comparison principle, large deviations were derived for the generalized Gibbs ensembles related with Toda lattices, allowing to describe their equilibrium measures in great generality, in terms of the equilibrium measures appearing in random matrix theory as proposed by Spohn. Moreover, the analysis of intricate high dimensional integrals similar to those arising in random matrices could be achieved and led to grounbreaking results in theoretical physics. First, it resulted with substantial progress in the description of two-point correlation functions for the 1+1 dimensional Sinh-Gordon quantum field theory. Second, it is conjectured in physics that solutions to systems of Bethe Ansatz equations exhibit condensation properties in the thermodynamic limit, similar to the eigenvalues of large random matrices. These properties have been established rigorously in the first part of the project for certain instances of Bethe Ansatz equations.

Multi-matrix integrals describe many models of physics and combinatorics since the work of ‘t Hooft and Brézin-Itzykson-Parisi-Zuber. Unfortunately, they are themselves very hard to compute or even estimate in general. On the way to understand such multi-matrix models, we first analyzed the simplest of them, namely spherical integrals. The asymptotics of spherical integrals were obtained for all range of ranks. This provided a key tool to establish large deviations principles for the spectrum of random matrices. Rectangular spherical integrals could also be estimated thanks to a new analysis of the asymptotics of Dyson Brownian bridges and more generally Brownian motion in mean field singular interactions. Besides these specific models, the general study of multi-matrix Gaussian integrals at high temperature was complemented by a topological expansion for multi-unitary matrix integrals. The study of multi-matrix models at low temperature was initiated, opening the door to a new understanding of the so-called commutator model.

Large deviation for the spectrum of large random matrices will be understood in great generality, both for the extremes eigenvalues and for the empirical measure of the eigenvalues. Over the next period, the most challenging question in this direction is related to the large deviation for the empirical measure of subgaussian entries.

Many new particles models with mean field singular interaction will be analyzed, much beyond the state of the art and Coulomb gases. Generalized Gibbs ensembles related with integrable systems will be fully understood, shading a new light on the equilibrium state of the Toda lattice.The understanding of the asymptotic analysis of singular integrals will be generalized in such extent to fully analyze the well known Sinh-interaction integrals describing correlation functions within the quantum separation of variables approach. Over the next period, we hope to be able to analyze the expressions for all multi-point correlation functions in the Sinh and Sine-Gordon quantum field theories and establish the well-definiteness of their series representations. Solving this question will allow to explicitly construct the full correlation function content of these theories by rigorously checking the validity of the Wightman axioms.

This will provide a fully explicit and rigorous construction of these quantum field theories deep in the interacting regime, which is a quite exceptional result taken the state of the art of the field.

New multi-matrix models will be analyzed, paving the way to a better understanding of Voiculescu's entropy.

Random matrix theory and large deviations theory will shade new lights in statistical and machine learning. In particular non Bayes optimal inference will be rigorously analyzed in great generality. Furthermore, new matrix models appearing in statistical learning will be analyzed, and new transition phenomena and algorithms established. Moreover, the analysis of critical points and minima of random functions arising in physics and machine learning will be extended to well known tensor models and their phase transition will be studied, much beyond the state of the art. These new developments will be based on new large deviations estimates for the spectrum of certain random matrices.