Spectrum of random d‐regular graphs up to the edge

Abstract

Consider the normalized adjacency matrices of random d‐regular graphs on N vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, . (ii) All eigenvectors of random d‐regular graphs are completely delocalized.

Directional extremal statistics for Ginibre eigenvalues

“…This largely completes the theory of Hermitian universality for the simplest mean field ensembles. However, much more work has been done and yet to be done on more general Hermitian ensembles, especially random band matrices [113,114,45,46,22,33,36,39,63,76,109,115,126,127,128,129], on adjacency matrices of sparse random graphs [82,83,23,84,10,11,12,13,80], and on Lévy matrices [4,5,6,20,25,29,30], that we will not discuss here. Non-Hermitian matrices exhibit similar universality patterns; although their mathematical theory is much less developed, the progress spectacularly accelerated in the last years.…”