Analytic Methods for the Random Matrix Universality Class

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The investigator will study random matrices, a field originating with Eugene Wigner's idea that random matrices model spectral behavior of physical systems. Over the years random matrices have found applications beyond pure mathematics, and are being used in statistics, computer science, telecommunications, and more generally information networks. This research project's focus is universality -- the phenomenon that the limiting spectral statistics depend only on the symmetry type and not on other details of the underlying system. Random matrix statistics now find use in many aspects of integrable systems, growth models, and number theory. Deep progress has been achieved in the past years, and universality has been established for a growing class of random matrices. This project will advance understanding in this fundamental area.This project explores the following research topics:(1) Universality and quantum unique ergodicity for random band matrices. The motivation is to try to approach the Anderson transition via these toy models for random Schrodinger operators on a lattice.(2) Perturbative analysis of eigenvectors in a non-perturbative regime, via the eigenvector moment flow, a new random walk introduced recently. (3) Log-correlated fields and random spectra. This includes individual fluctuations of eigenvalues of Wigner matrices and beta ensembles. (4) Extremal statistics of random matrices, through the largest and smallest gaps, and extremes of characteristic polynomials.(5) A study of non-Hermitian random matrix theory, with connections between 2D Coulomb gases and the Gaussian free field.In order to understand Wigner's vision, in their recent proofs of fixed energy universality and eigenvector universality the investigator and collaborators developed new tools of interest for the above projects. These include random walks in dynamic random environments, coupling methods, and homogenization theory for partial differential equations with time-dependent random coefficients.