Ramanujan Property and Edge Universality of Random Regular Graphs

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\ge 3$ and denote its eigenvalues as ${\lambda }_1=d\mathrm{/}\sqrt{d-1}\ge {\lambda }_2\ge {\lambda }_3\cdots \ge {\lambda }_N$. We establish the following two results as $N\to \mathrm{\infty }$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{\mathrm{o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+\mathrm{o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2\mathrm{/}3+\mathrm{o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of ${\lambda }_2$ and $-{\lambda }_N$ converge to the Tracy-Widom${}_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\mathrm{\%}$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max \{{\lambda }_2,\mathrm{\mid }{\lambda }_N\mathrm{\mid }\}\le 2$.