Random perfect matchings in regular graphs

We prove that in all regular robust expanders G$$ G $$, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching M$$ M $$. We also show that given any fixed matching or spanning regular graph N$$ N $$ in G$$ G $$, the random variable |M & AND;E(N)|$$ \mid M\cap E(N)\mid $$ 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.