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The second largest component in the supercritical 2D Hamming graph

By Malwina Luczak and J Spencer

Abstract

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 ln1/3n << 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

Topics: H Social Sciences (General)
Publisher: London school of economics and political science
Year: 2007
OAI identifier: oai:eprints.lse.ac.uk:6808
Provided by: LSE Research Online
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