Random perfect matchings in regular graphs

Abstract

AbstractWe prove that in all regular robust expanders , every edge is asymptotically equally likely contained in a uniformly chosen perfect matching . We also show that given any fixed matching or spanning regular graph in , the random variable 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.