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

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.

The draft is improved: Typos are fixed, Corollaries 4.3 and 4.5 are improved, and Proposition 4.6 and Remark 4.7 are added