Periodic Walks on Large Regular Graphs and Random Matrix Theory

TL;DR

This paper investigates the distribution of non-backtracking periodic walks on large regular graphs, proposing a formula relating variance and mean based on Random Matrix Theory, supported by numerical and theoretical evidence.

Contribution

It introduces a conjecture linking spectral statistics of regular graphs to RMT and provides a trace formula connecting spectral density with periodic walks.

Findings

Proposes a formula for variance-to-mean ratio of periodic walks.

Provides numerical and theoretical support for the RMT spectral conjecture.

Connects spectral density to periodic walks via a trace formula.

Abstract

We study the distribution of the number of (non-backtracking) periodic walks on large regular graphs. We propose a formula for the ratio between the variance of the number of $t$-periodic walks and its mean, when the cardinality of the vertex set $V$ and the period $t$ approach $\infty$ with $t/V\rightarrow \tau$ for any $\tau$. This formula is based on the conjecture that the spectral statistics of the adjacency eigenvalues is given by Random Matrix Theory (RMT). We provide numerical and theoretical evidence for the validity of this conjecture. The key tool used in this study is a trace formula which expresses the spectral density of $d$-regular graphs, in terms of periodic walks.