Spectral gap and edge universality of dense random regular graphs

Let $\mathcal{A}$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $\lambda_1\geq \lambda_2\cdots \geq \lambda_{N}$\mathrm{.}For$N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of the extreme eigenvalues of $\mathcal{A}$, which in particular imply that $$\max \{\mathrm{\mid }{\lambda }_N\mathrm{\mid },{\lambda }_2\}<2\sqrt{d-1}$$ with overwhelming probability. In the same regime of $d$, we also show that $$N^{2\mathrm{/}3}(\frac{{\lambda }_2+d\mathrm{/}N}{\sqrt{d(N-d)\mathrm{/}N}}-2){\mathrm{T}\mathrm{W}}_1,$$where ${\mathrm{T}\mathrm{W}}_1$ is the Tracy-Widom distribution for GOE; analogues results also hold for other non-trivial extreme eigenvalues.