# Optimal Eigenvalue Rigidity of Random Regular Graphs

arxiv（2024）

Abstract

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.

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