The number of perfect matchings, and the nesting properties, of random regular graphs

Abstract

We prove that the number of perfect matchings in 𝒢false(n,dfalse) is asymptotically normal when n$$ n $$ is even, d→∞$$ d\to \infty $$ as n→∞$$ n\to \infty $$, and d=Ofalse(n1false/7false/lognfalse)$$ d=O\left({n}^{1/7}/\log n\right) $$. This is the first distributional result of spanning subgraphs of 𝒢false(n,dfalse) when d→∞$$ d\to \infty $$. Moreover, we prove that 𝒢false(n,dprefix−1false) and 𝒢false(n,dfalse) can be coupled so that 𝒢false(n,dprefix−1false) is a subgraph of 𝒢false(n,dfalse) with high probability when d→∞$$ d\to \infty $$ and d=ofalse(n1false/3false)$$ d=o\left({n}^{1/3}\right) $$. Furthermore, if d=ωfalse(log7nfalse)$$ d=\omega \left({\log}^7n\right) $$, d=Ofalse(n1false/7false/lognfalse)$$ d=O\left({n}^{1/7}/\log n\right) $$, and d≤d′≤nprefix−1$$ d\le {d}^{\prime}\le n-1 $$ then 𝒢false(n,dfalse) and 𝒢false(n,d′false) can be coupled so that asymptotically almost surely (a.a.s.) 𝒢false(n,dfalse) is a subgraph of 𝒢false(n,d′false).