Local Limit Theorems and Number of Connected Hypergraphs

Details Local Limit Theorems and Number of Connected Hypergraphs Local Limit Theorems and Number of Connected Hypergraphs

2007-06-04

arxiv.org/abs/0706.0497v1

Mathematics

Combinatorics

24 pages

Scientific paper

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

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