Eigenvalue fluctuations for random regular graphs

Abstract

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying the spectral properties of random regular graphs. Specifically, we prove limit theorems for the fluctuations of linear spectral statistics of random regular graphs. We find both universal and non-universal behavior. Our most important tool is Stein's method for Poisson approximation, which we develop for use on random regular graphs.