The second largest component in the supercritical 2D Hamming graph

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.

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Remco van der Hofstad, Malwina J. Luczak, Joel Spencer
• Keywords: 2-dimensional Hamming Graph, Largest Component, Random Subgraphs, RSA 2010