Spectral gap and edge universality of dense random regular graphs

Abstract

Let $\mathcal{A}$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by ${\lambda }_1\ge {\lambda }_2\cdots \ge {\lambda }_N$. For $N^{2/3}\ll d\le N/2$, we prove optimal rigidity estimates of the extreme eigenvalues of $\mathcal{A}$, which in particular imply that \[ \max\{|λ_N|,λ_2\} <2\sqrt{d-1} \] with overwhelming probability. In the same regime of $d$, we also show that \[ N^{2/3}\bigg(\frac{λ_2+d/N}{\sqrt{d(N-d)/N}}-2\bigg) \overset{d}{\longrightarrow} \mathrm{TW}_1\,, \]where ${\mathrm{TW}}_1$ is the Tracy-Widom distribution for GOE; analogues results also hold for other non-trivial extreme eigenvalues.

34 pages

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