Work

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

Year: 2025

Type: article

Abstract: In this paper, we 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 ‐regular graphs with vertices, if and tends to infinity, the normalized spectral measure converges almost surely in ‐Wasserstein distance to the semicircle distribution for any . This strengthens a result of Dumitriu and Pal. Many of the results are also extended to unitary‐colored regular graphs. For example, we give a short proof of the weak convergence to the Kesten–McKay distribution for the normalized spectral measures of random ‐lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non‐backtracking walks.

Source: Mathematische Nachrichten

DOI: 10.1002/mana.70046

Authors: Yulin Gong, Wenbo Li, Shiping Liu

Institutions: Tsinghua University, University of Science and Technology of China

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, National Key Research and Development Program of China

Awards: G1937537291, G2904574553, G3718149727, G5641300969, G6759021320