Edge universality of random regular graphs of growing degrees

We consider the statistics of extreme eigenvalues of random d-regular graphs, with Nc≤d≤N1/3−c for arbitrarily small 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 d−1.