Abstract
We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum {[-2sqrt{d-1}+varepsilon, 2sqrt{d-1}-varepsilon]} down to the optimal spectral scale, we prove that the Green’s functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten–McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green’s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs.
Highlights
Random regular graphs with fixed degree d are fundamental models of sparse random graphs and they arise naturally in many different contexts
Macroscopic eigenvalue statistics for random regular graphs of fixed degree have been studied using the techniques of Poisson approximation of short cycles [31,52] and using the replica method [66]. These results show that the macroscopic eigenvalue statistics for random regular graphs of fixed degree are different from those of a Gaussian matrix
For any graph G, the adjacency matrix is the symmetric matrix A indexed by the vertices of the graph, with Ai j = A ji = 1 if there is an edge between i and j, and Ai j = 0 ot√herwise
Summary
Random regular graphs with fixed degree d are fundamental models of sparse random graphs and they arise naturally in many different contexts. There has been significant progress in the understanding of the spectra of (random and deterministic) regular graphs. For fixed degree these results generally concern properties of eigenvalues and eigenvectors near the macroscopic scale, and their proofs use the local tree-like structure of these graphs as an important input. Dense regular graphs belong to the random matrix universality class and their spectral properties are known to resemble those of Wigner matrices. We introduce an approach that allows the Green’s function method of random matrix theory to make use of the local tree-like structure of the random regular graph, while it captures key random matrix behavior
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