Optimal Eigenvalue Rigidity of Random Regular Graphs

TL;DR

This paper proves optimal eigenvalue rigidity for random $d$-regular graphs, showing eigenvalues are tightly concentrated around their classical locations with fluctuations matching those of GOE matrices.

Contribution

It establishes the first optimal eigenvalue rigidity bounds for random regular graphs, matching the fluctuation scale of Gaussian ensembles.

Findings

Eigenvalues are within $N^{o(1)}/N^{2/3}$ of their classical locations.

Extreme eigenvalues fluctuate on the scale of $N^{-2/3+o(1)}$.

Eigenvalue fluctuations match those of Gaussian Orthogonal Ensemble.

Abstract

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We prove that the optimal (up to an extra $N^{{\rm o}_N(1)}$ factor, where ${\rm o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote $\gamma_i$ as the classical location of the $i$-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\rm o}_N(1)}$, \begin{align*} |\lambda_i-\gamma_i|\leq \frac{N^{{\rm o}_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \text{ for all } i\in \{2,3,\cdots,N\}. \end{align*} In particular, the fluctuations of extreme eigenvalues are bounded by $N^{-2/3+{\rm o}_N(1)}$. This gives the same order of fluctuation as for the eigenvalues of matrices from the Gaussian Orthogonal Ensemble.