We prove that the number of perfect matchings in ${ \mathcal G}(n,d)$ is asymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and $d=O(n^{ 1/7}/\log^2 n)$. This is the first distributional result of spanning subgraphs of ${ \mathcal G}(n,d)$ when $d\to\infty$. Moreover, we prove that ${ \mathcal G}(n,d-1)$ and ${ \mathcal G}(n,d)$ can be coupled so that ${ \mathcal G}(n,d-1)$ is a subgraph of ${ \mathcal G}(n,d)$ with high probability when $d\to\infty$ and $d=o(n^{ 1/3})$. Further, if $d=\Omega(\log^7 n)$, $d=O(n^{ 1/7}/\log^2n)$, and $d\le d'\le n-1$ then ${ \mathcal G}(n,d)$ and ${ \mathcal G}(n,d')$ can be coupled so that asymptotically almost surely ${ \mathcal G}(n,d)$ is a subgraph of ${ \mathcal G}(n,d')$.
Categories: math.CO