Local Limit Theorems and Number of Connected Hypergraphs

Let Hd(n, p) signify a random d-uniform hypergraph with n vertices in which each of the ( n d ) possible edges is present with probability p = p(n) independently, and let Hd(n, m) denote a uniformly distributed with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of Hd(n, p) and Hd(n, m) for the regime ( n−1 d−1 ) p, dm/n > (d−1) + ε. As an application, we obtain an asymptotic formula for the probability that Hd(n, p) or Hd(n, m) is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in Hd(n, p) given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.