CDAM: Computational, Discrete and Applicable Mathematics@LSE

CDAM Research Report, LSE-CDAM-2007-39

February 2008

The Second Largest Component in the Supercritical 2D Hamming Graph

Malwina J. Luczak and Joel Spencer

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 ln^1/3n << 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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		CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K.
		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk