Local Kesten–McKay Law for Random Regular Graphs

Roland Bauerschmidt, Jiaoyang Huang, Horng Tzer Yau

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)523-636
Number of pages114
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jul 1 2019

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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