Random matrices are models for disordered physical systems, describing key properties of their energy levels or eigenstates. This probabilistic field now overlaps many aspects of integrable systems, growth models, number theory, or multivariate statistics. The mathematical understanding of these models has considerably improved over the past ten years, based on analytic methods for invariant models, and probabilistic methods for models with underlying independence, including the analysis of Dyson's Brownian motion. This research project first concerns the local universality of such statistics, enlarging the class of models presenting the random matrices type of interactions; this includes the study of the so-called beta-ensembles, proving that the local interactions of the energy levels of many random Hamiltonian systems only depend on the their invariance type, and the analysis of universality for bidimensional spectra, appearing in non-Hermitian random matrix theory. Another part of this project concerns the mesoscopic scale in random matrix theory, deriving new statistics relevant in analytic number theory and random energy models.
Indeed, surprisingly random matrix theory overlaps fundamental problems in analytic number theory, statistical physics and statistics. Analogously to the Gaussian distribution, describing the fluctuations of many systems with underlying independence, random matrix theory statistics appear universally for many strongly correlated systems. This as confirmed by physical and numerical experiments concerning the energy levels of quantum systems, waiting times in public transports, the gap size distribution of parked cars, growing interfaces of liquid crystal turbulence, or typical and extreme spacings between zeros of L-functions. Random matrices are paradigms for statistics appearing in very distinct fields, and an increasingly important part of probability theory is devoted to understanding these connections.
