Random Matrix Theory
Summer School
in Japan 2025

Alice Guionnet (ENS Lyon)

Large deviations for the largest eigenvalue of large random matrices, and applications.

Abstract: In this mini-course I will review several recent large deviation results for large random matrices, and their relations with spin glasses and the study of the volume of local minima of random functions. This is based on joint works with G. Ben Arous, J. Boursier, N. Cook, A. Dembo, R. Ducatez, J. Husson, and M. Maida.

Andrea Nahmod (University of Massachusetts Amherst)

TBA

László Erdős (IST Austria)

Multi-resolvent local laws and their applications

Abstract: Classical local laws in random matrix theory assert that the resolvents of large random matrices tend to be deterministic even for spectral parameters very close to the real axis. They are robust and provide essential a priori bounds for eigenvalue and eigenvector distributions that are routinely used in more sophisticated analysis. Products of resolvents also tend to be deterministic, but they are not simply given as a product of single resolvent approximations. In this series of lectures we present a theory of multi-resolvent local laws. The proofs are dynamical, they rely on the so-called zig-zag strategy; a successive alternate application of two different stochastic flows. In the second part of the lectures we focus on applications that include the proof of the Eigenstate Thermalisation Hypothesis and the Law of Fractional Logarithm for general Hermitian random matrices, as well as CLT for linear eigenvalue statistics and the Gumbel law for extremal eigenvalues for non-Hermitian random matrices.