Exploring hypergraphs with martingales

Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper "Counting connected hypergraphs via the probabilistic method" to enumerate sparsely connected hypergraphs asymptotically.Comment: 32 pages; significantly expanded presentation. To appear in Random Structures and Algorithm

Counting connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity $m-n+1$ tend to infinity. Asymptotic formulae for the number of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges and so nullity $t=(r-1)m-n+1$ were proved by Karo\'nski and \L uczak for the case $t=o(\log n/\log\log n)$, and Behrisch, Coja-Oghlan and Kang for $t=\Theta(n)$. Here we prove such a formula for any $r\ge 3$ fixed, and any $t=t(n)$ satisfying $t=o(n)$ and $t\to\infty$ as $n\to\infty$. This leaves open only the (much simpler) case $t/n\to\infty$, which we will consider in future work. ( arXiv:1511.04739 ) Our approach is probabilistic. Let $H^r_{n,p}$ denote the random $r$-uniform hypergraph on $[n]$ in which each edge is present independently with probability $p$. Let $L_1$ and $M_1$ be the numbers of vertices and edges in the largest component of $H^r_{n,p}$. We prove a local limit theorem giving an asymptotic formula for the probability that $L_1$ and $M_1$ take any given pair of values within the `typical' range, for any $p=p(n)$ in the supercritical regime, i.e., when $p=p(n)=(1+\epsilon(n))(r-2)!n^{-r+1}$ where $\epsilon^3n\to\infty$ and $\epsilon\to 0$; our enumerative result then follows easily. Taking as a starting point the recent joint central limit theorem for $L_1$ and $M_1$, we use smoothing techniques to show that `nearby' pairs of values arise with about the same probability, leading to the local limit theorem. Behrisch et al used similar ideas in a very different way, that does not seem to work in our setting. Independently, Sato and Wormald have recently proved the special case $r=3$, with an additional restriction on $t$. They use complementary, more enumerative methods, which seem to have a more limited scope, but to give additional information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical changes. 67 pages (including appendix