Spectral convergence of random regular graphs: Chebyshev polynomials, non-backtracking walks, and unitary-color extensions

TL;DR

This paper proves the spectral measure convergence of random regular graphs to the Kesten-McKay and semicircle distributions, extending previous results with new proofs and generalizations involving Chebyshev polynomials, non-backtracking walks, and unitary-color extensions.

Contribution

It provides a simplified proof of spectral convergence for random lifts and extends criteria for spectral measure convergence to regular graphs with growing degree.

Findings

Spectral measures of random regular graphs converge to the Kesten-McKay distribution.

Normalized spectral measures of large regular graphs tend to the semicircle distribution.

Results are extended to unitary-colored regular graphs.

Abstract

In this paper, we give a short proof of the weak convergence to the Kesten-McKay distribution for the normalized spectral measures of random $N$-lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non-backtracking walks. We also extend a criterion of Sodin on the convergence of graph spectral measures to regular graphs of growing degree. As a result, we show that for a sequence of random $(q_n+1)$-regular graphs $G_n$ with $n$ vertices, if $q_n = n^{o(1)}$ and $q_n$ tends to infinity, the normalized spectral measure converges almost surely in $p$-Wasserstein distance to the semicircle distribution for any $p \in [1, \infty)$. This strengthens a result of Dumitriu and Pal. Many of the results are extended to unitary-colored regular graphs.