Edge Universality of Random Regular Graphs of Growing Degrees

TL;DR

This paper proves that the extreme eigenvalues of random $d$-regular graphs with growing degrees follow the Tracy-Widom distribution, establishing edge universality in this regime.

Contribution

It demonstrates edge universality for the extreme eigenvalues of random regular graphs with degrees growing polynomially with the number of vertices.

Findings

Extreme eigenvalues follow Tracy-Widom distribution

Approximately 69% of graphs have eigenvalues within the spectral bound

Edge universality holds for degrees up to $N^{1/3-\mathfrak{c}}$

Abstract

We consider the statistics of extreme eigenvalues of random $d$-regular graphs, with $N^{\mathfrak c}\leq d\leq N^{1/3-{\mathfrak c}}$ for arbitrarily small ${\mathfrak c}>0$. We prove that in this regime, the fluctuations of extreme eigenvalues are given by the Tracy-Widom distribution. As a consequence, about 69% of $d$-regular graphs have all nontrivial eigenvalues bounded in absolute value by $2\sqrt{d-1}$.