Random perfect matchings in regular graphs

TL;DR

This paper demonstrates that in regular robust expanders, each edge is nearly equally likely to be in a random perfect matching, and the intersection with fixed matchings follows a Poisson distribution, confirming conjectures and extending prior results.

Contribution

It proves uniform edge probabilities in perfect matchings of regular robust expanders and characterizes intersection distributions, advancing understanding of matchings in such graphs.

Findings

Edges are asymptotically equally likely in perfect matchings.

Intersection sizes with fixed matchings are approximately Poisson distributed.

Confirms a conjecture by Spiro and Surya.

Abstract

We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.