The second largest component in the supercritical 2D Hamming graph

Abstract

AbstractThe two‐dimensional Hamming graph H(2,n) consists of the n2 vertices (i,j), 1 ≤ i,j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, in such a way that the average degree satisfies 2(n − 1)p = 1 + ε. Previous work [8] has shown that in the barely supercritical region n−2/3 ln1/3n ≪ ε ≪ 1, the largest component satisfies a law of large numbers with mean 2εn. Here we show that the second largest component has, with high probability, size bounded by 28ε−2 log(n2ε3), so that the dominant component has emerged. This result also suggests that a discrete duality principle holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010