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 $d⩾3$. We prove that, with probability $1-N^{-1+\epsilon }$ for any $\epsilon >0$, the following two properties hold as $N\to \infty$ provided that $d⩾3$: (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, ${\lambda }_2,\left|{\lambda }_N\right|⩽2+N^{-c}$. (ii) All eigenvectors of random d-regular graphs are completely delocalized.