Ramanujan Property and Edge Universality of Random Regular Graphs

Understand

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 $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$.

• 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 $N^{-2/3+{\rm o}(1)}$.