Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)

TL;DR

This paper links spectral statistics of large random regular graphs to Random Matrix Theory predictions, showing that cycle counts follow Poisson distribution and deviations can be computed assuming RMT correlations.

Contribution

It establishes a connection between spectral statistics of regular graphs and RMT, extending the trace formulae to analyze cycle count distributions and their deviations.

Findings

Cycle counts follow Poisson distribution in large regular graphs.

Spectral correlations align with RMT predictions to all orders.

Numerical evidence supports the conjecture of RMT-based deviations.

Abstract

Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from the known Poisson distribution of cycle counts in regular graphs, in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely. The result is analogous to the so called "diagonal approximation" in Quantum Chaos. We also show that by assuming that the spectral correlations are given by RMT to all orders, we can compute the leading deviations from the Poisson distribution for cycle counts. We provide numerical evidence which supports this conjecture.