Local limit theorems for the giant component of random hypergraphs

Let H-d (n, p) signify a random d-uniform hypergraph with n vertices in which each of the ((n)(d)) possible edges is present with probability d p = p(n) independently, and let Hd(n, m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We establish a local limit theorem for the number of vertices and edges in the largest component of Hd (n, p) in the regime (d - 1)(-1) + epsilon < ((n-1)(d-1)) p = O(1), thereby determining the joint distribution of these parameters precisely. As an application, we derive an asymptotic formula for the probability that Hd (n, m) is connected, thus obtaining a formula for the asymptotic number of connected hypergraphs with a given number of vertices and edges. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.