Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2006-15

October 2006

Random Subgraphs of the 2D Hamming Graph: The Supercritical Phase

Remco van der Hofstad and Malwina J. Luczak

We study random subgraphs of the 2-dimensional Hamming graph $H(2,n)$, which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In \cite{bchss1, bchss2}, the size of the largest connected component was estimated precisely for a large class of graphs including $H(2,n)$ for $\vep\leq \Lambda V^{-1/3}$, where $\Lambda > 0$ is a constant and $V=n^2$ denotes the number of vertices in $H(2,n)$. Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when $\vep\gg (\log{V})^{1/3} V^{-1/3}$, then the largest connected component has size close to $2\vep V$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ are supercritical. Barring the factor $(\log{n})^{1/3}$, this identifies the size of the largest connected component all the way down to the critical $p$ window.

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