Dynamical Universality for Random Matrices

Abstract: We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove the dynamical universality of random matrices in the sense that, if the random point fields $ \muN $ of $ \nN $-particle systems describing the eigenvalues of random matrices or log-gases with general self-interaction potentials $ \V $ converge to some random point field $ \mu $, then the associated natural $ \muN $-reversible diffusions represented by solutions of stochastic differential equations (SDEs) converge to some $ \mu $-reversible diffusion given by the solution of an infinite-dimensional SDE (ISDE). % Our results are general theorems that can be applied to various random point fields related to random matrices such as sine, Airy, Bessel, and Ginibre random point fields. % In general, the representations of finite-dimensional SDEs describing $ \nN $-particle systems are very complicated. Nevertheless, the limit ISDE has a simple and universal representation that depends on a class of random matrices appearing in the bulk, and at the soft- and at hard-edge positions. Thus, we prove that ISDEs such as the infinite-dimensional Dyson model and the Airy, Bessel, and Ginibre interacting Brownian motions are universal dynamical objects.