  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 ${\mathcal
G}(n,d')$.

Combinatorics