Optimal Eigenvalue Rigidity of Random Regular Graphs

Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree d≥ 3, and denote the eigenvalues as λ_1=d/√(d-1)≥λ_2≥λ_3⋯≥λ_N. We prove that the optimal (up to an extra N^ o_N(1) factor, where o_N(1) can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote γ_i as the classical location of the i-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability 1-N^-1+ o_N(1), |λ_i-γ_i|≤N^ o_N(1)/N^2/3 (min{i,N-i+1})^1/3, for all i∈{2,3,⋯,N}. In particular, the fluctuations of extreme eigenvalues are bounded by N^-2/3+ o_N(1). This gives the same order of fluctuation as for the eigenvalues of matrices from the Gaussian Orthogonal Ensemble.