Local Limit Theorems for the Giant Component of Random Hypergraphs

Abstract

Let Hd(n,p) signify a random d-uniform hypergraph with n vertices in which each of the $\binom{n}{d}$ possible edges is present with probability p=p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph 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) in the regime $(d-1)\binom{n-1}{d-1}p>1+\varepsilon$, resp. d(d−1)m/n>1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach.