Eigenvalue fluctuations for random regular graphs

TL;DR

This paper investigates the spectral fluctuations of random regular graphs, revealing both universal and non-universal behaviors, and introduces Stein's method for Poisson approximation tailored to these graphs.

Contribution

It proves limit theorems for spectral fluctuations of random regular graphs and develops Stein's method for Poisson approximation in this context.

Findings

Identifies universal and non-universal spectral fluctuation behaviors.

Develops Stein's method for Poisson approximation on regular graphs.

Provides detailed proofs and improvements over previous work.

Abstract

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. 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. This is my Ph.D. thesis, based on joint work with Ioana Dumitriu, Elliot Paquette, and Soumik Pal. For the most part, it's a mashed up version of arXiv:1109.4094, arXiv:1112.0704, and arXiv:1203.1113, but some things in here are improved or new. In particular, Chapter 4 goes into more detail on some of the proofs than arXiv:1203.1113 and includes a new section. See Section 1.3 for more discussion on what's new and who contributed to what.