CDAM Research Report, LSE-CDAM-2007-39
The Second Largest Component in the Supercritical 2D Hamming Graph
Malwina J. Luczak and Joel SpencerThe 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  had shown that in the barely supercritical region n-2/3 ln1/3n << ε << 1 the largest component has size ~ 2εn. Here we show that the second largest component has size close to ε-2, so that the dominant component has emerged.
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