Local Limit Theorems and Number of Connected Hypergraphs

Abstract

Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the $\left(\genfrac{}{}{0ex}{}{n}{d}\right)$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ 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 $H_d(n,p)$ and $H_d(n,m)$ for the regime $\left(\genfrac{}{}{0ex}{}{n-1}{d-1}\right)p,dm/n>(d-1{)}^{-1}+\epsilon$. As an application, we obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in $H_d(n,p)$ given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.

24 pages

References in corpus (1)

• Creation and Growth of Components in a Random Hypergraph Process

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