The 2-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, so that the average degree 2(n−1)p = 1 + ε. Previous work [5] had shown that in the barely supercritical region n−2/3 ln1/3 n ε 1 the largest component has size ∼ 2εn. Here we show that the second largest component has size bounded by ε−2 log(nε3 ), 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.
Remco van der Hofstad, Malwina J. Luczak, Joel Spe