EDGE RIGIDITY AND UNIVERSALITY OF RANDOM REGULAR GRAPHS OF INTERMEDIATE DEGREE

For random d-regular graphs on N vertices with 1 << d << N-2/3, we develop a d(-1/2) expansion of the local eigenvalue distribution about the Kesten-McKay law up to order d(-3). This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erdos-Renyi graphs of the same average degree. As a first application, for 1 << d << N-2/3, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by (2 + o(1))root d - 1. As a second application, for N-2/9 << d << N-1/3, we prove that the extremal eigenvalues are concentrated at scale N-2/3 and their fluctuations are governed by Tracy-Widom statistics. Thus, in the same regime of d, 52% of all d-regular graphs have second-largest eigenvalue strictly less than 2 root d - 1.