Random subgraphs of the 2D Hamming graph: the supercritical phase

Abstract

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 \({p={(1+\varepsilon)}/{(2(n-1))}}\) for some \({\varepsilon \in \mathbb{R}}\) . In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for \({\varepsilon \leq \Lambda V^{-1/3}}\) , where Λ > 0 is a constant and V = n2 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 \({\varepsilon \gg (\log{V})^{1/3} V^{-1/3}}\) , then the largest connected component has size close to \({2 \varepsilon V}\) 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 \({(\log{V})^{1/3}}\), this identifies the size of the largest connected component all the way down to the critical p window.