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July 2004 Percolation on finite graphs and isoperimetric inequalities
Noga Alon, Itai Benjamini, Alan Stacey
Ann. Probab. 32(3): 1727-1745 (July 2004). DOI: 10.1214/009117904000000414

Abstract

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.

Citation

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Noga Alon. Itai Benjamini. Alan Stacey. "Percolation on finite graphs and isoperimetric inequalities." Ann. Probab. 32 (3) 1727 - 1745, July 2004. https://doi.org/10.1214/009117904000000414

Information

Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1046.05071
MathSciNet: MR2073175
Digital Object Identifier: 10.1214/009117904000000414

Subjects:
Primary: 05C80, 60K35

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • July 2004
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