Edge universality for non-trivial random regular graphs

Let $G$ be a uniformly random $d$-regular graph on $N$ vertices, with adjacency eigenvalues $\boldsymbol{\textstyle\frac{d(N-1)}{N}}$ excluded from consideration, and let $\boldsymbol{\lambda_2}$ denote its second eigenvalue. Assume $3\leqslant d\leqslant N-4$. The notation $\mathrm{TW}_1$ denotes the Tracy–Widom distribution for GOE.

Edge universality conjecture. We have

$$N^{2/3}\bigg(\frac{\lambda_2}{\sqrt{(d-1)(N-d-2)/N}}-2\bigg) \overset{d}{\longrightarrow} \mathrm{TW}_1,$$

where analogous results also hold for other non-trivial extreme eigenvalues.

The conjecture proposes edge universality for all non-trivial random $d$-regular graphs, extending the proved dense regime and respecting the degree symmetry $d\longleftrightarrow N-d-1$. The authors indicate that the range $d\leqslant N^{2/9}$, especially fixed $d$, is difficult, while suggesting that optimal edge rigidity and universality may be provable for $N^{2/9}\ll d\leqslant N/2$.