Local spectral stability for random regular graphs of fixed degree

Roland Bauerschmidt, Jiaoyang Huang, Horng-Tzer Yau

Published 2016-09-28Version 1

We study random $d$-regular graphs with large but fixed degree $d$. In the bulk spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$, we prove that, while its entries do not concentrate, the Green's function is accurately approximated by random variables that only depend on the local structure of the graph, down to the optimal spectral scale. This stability result implies, among other consequences, that the Kesten--McKay law for the spectral density applies down to the smallest possible scale, and that the bulk eigenvectors are completely delocalized and satisfy a strong form of quantum unique ergodicity. Our method combines the well-known tree-like (few cycles) structure of random regular graphs at small distances with random matrix-like behavior at large distances.