Spectral Gap and Edge Universality of Dense Random Regular Graphs

Let A be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by lambda(1) >= lambda(2) . . . >= lambda(N). For N2/3+o(1) <= d <= N/2, we prove optimal rigidity estimates of the extreme eigenvalues of A, which in particular imply that max{|lambda(N)|, lambda(2)} < 2 root d-1 with very high probability. In the same regime of d, we also show that N-2/3(lambda(2)+d/N root d(N-d)/N-2) (sic) TW1, where TW1 is the Tracy-Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues.