Local Limit Theorems for the Giant Component of Random Hypergraphs

Abstract

LetHd(n,p)signify a randomd-uniform hypergraph withnvertices in which each of the$\binom{n}{d}$possible edges is present with probabilityp=p(n)independently, and letHd(n,m)denote a uniformly distributedd-uniform hypergraph withnvertices andmedges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component ofHd(n,p)andHd(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 asn→ ∞. The proofs are based on a purely probabilistic approach.