Subgraph distributions in dense random regular graphs

Given connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of length $3,4,5$ as well as paths of length $3$ in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $\mathbb{G}(n,p)$

Degree sequences of sufficiently dense random uniform hypergraphs

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$-uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum