Edge Universality of Random Regular Graphs of Growing Degrees

Abstract

We consider the statistics of extreme eigenvalues of random $d$-regular graphs, with $N^{\mathfrak{c}}\le d\le N^{1/3-\mathfrak{c}}$ 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 $2\sqrt{d-1}$.

53 pages, 3 figures