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

We prove that the number of perfect matchings in G(n, d) is asymptotically normal when n is even, d -> infinity as n -> infinity, and d = O(n(1/7)/ log n). This is the first distributional result of spanning subgraphs of G(n, d) when d -> infinity. Moreover, we prove tha G(n, d - 1) and G(n, d) can be coupled so that G(n, d) is a subgraph of G(n, d) with high probability when d -> infinity and d = o(n(1/3)). Furthermore, if d = omega(log(7) n) d = O(n(1/7) / log n), and d <= d' <= n - 1 then G(n, d) and G(n, d') can be coupled so that asymptotically almost surely (a.a.s.) G(n, d) is a subgraph of G(n, d').