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Random subgraphs of the 2D Hamming graph: the supercritical phase

van der Hofstad, R and Luczak, Malwina (2007) Random subgraphs of the 2D Hamming graph: the supercritical phase. LSE-CDAM-2007-37. London school of economics and political science, London, UK.

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Abstract

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p=(1+ε)/2(n-1) for some ε∈ R. In [4,5], the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for ε≤ Λ V-1/3, where Λ > 0 is a constant and V=n2 denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ε>> (log V)1/3 V-1/3, then the largest connected component has size close to 2ε V with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor (log V)1/3, this identifies the size of the largest connected component all the way down to the critical p window.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk/Reports/
Additional Information: © 2007 London school of economics and political science
Library of Congress subject classification: H Social Sciences > H Social Sciences (General)
Sets: Departments > Mathematics
Research centres and groups > Computational, Discrete and Applicable Mathematics@LSE (CDAM)
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: LSE-CDAM-2007-37
Date Deposited: 10 Jul 2008 09:16
URL: http://eprints.lse.ac.uk/6801/

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