Exchangeable pairs, switchings, and random regular graphs

TL;DR

This paper advances understanding of cycle counts in random regular graphs by extending asymptotic regimes, providing explicit bounds, and linking combinatorial switchings with Stein's method for spectral analysis.

Contribution

It introduces a novel application of Stein's method with exchangeable pairs to analyze cycle counts and spectral properties, connecting two different combinatorial techniques.

Findings

Cycle counts are approximately Poisson in a broader regime.

Explicit total variation bounds for cycle count approximations.

Derived limiting distributions for eigenvalue functionals.

Abstract

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue functionals for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.