The limit theorem with respect to the matrices on non-backtracking paths of a graph

TL;DR

This paper establishes a limit theorem for matrices associated with non-backtracking paths in regular graphs, revealing connections to the arcsine law and Fourier coefficients of cusp forms in Ramanujan graphs.

Contribution

It introduces a new limit theorem for matrices on non-backtracking paths and links it to spectral properties of Ramanujan graphs and cusp forms.

Findings

Limit theorem closely related to the arcsine law

Asymptotic behavior of Fourier coefficients of cusp forms

Connections between non-backtracking matrices and spectral graph theory

Abstract

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.