Edge rigidity and universality of random regular graphs of intermediate degree

TL;DR

This paper develops a detailed eigenvalue distribution expansion for random regular graphs with intermediate degree, showing they are more rigid than Erdős-Rényi graphs and analyzing their extremal eigenvalues with Tracy-Widom fluctuations.

Contribution

It provides a $d^{-1/2}$ expansion of the local eigenvalue distribution up to the spectrum edge and demonstrates eigenvalue rigidity and fluctuation results in specific degree regimes.

Findings

Eigenvalues are more rigid than in Erdős-Rényi graphs.

Nontrivial eigenvalues are bounded by $(2 + o(1)) \, \sqrt{d - 1}$ with high probability.

Extremal eigenvalues follow Tracy-Widom distribution in certain degree ranges.

Abstract

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$. This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erd\H{o}s-R\'enyi graphs of the same average degree. As a first application, for $1 \ll d \ll N^{2/3}$, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by $(2 + o(1)) \sqrt{d - 1}$. As a second application, for $N^{2/9} \ll d \ll N^{1/3}$, we prove that the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have second-largest eigenvalue strictly less than $2 \sqrt{d - 1}$.