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 + ε)/(2(n - 1)) for some ε ∈ 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 ε ≤ ΛV-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 ε » (log V)1/3V-1/3, then the largest connected component has size close to 2εV 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. © Springer-Verlag 2009.