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Local Kesten–McKay Law for Random Regular Graphs

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Huang, J 
Yau, HT 

Abstract

We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum [−2d−1+ε,2d−1−ε] 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.

Description

Keywords

math.PR, math.PR, math-ph, math.CO, math.MP, 05C80, 05C50, 60B20, 15B52

Journal Title

Communications in Mathematical Physics

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

369

Publisher

Springer Science and Business Media LLC