Ramanujan Property and Edge Universality of Random Regular Graphs

Abstract

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $λ_2$ and $-λ_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{λ_2,|λ_N|\}\leq 2$.