Percolation on finite graphs and isoperimetric inequalities

Percolation on finite graphs and isoperimetric inequalities

Noga Alon, Itai Benjamini, and Alan Stacey

Source: Ann. Probab. Volume 32, Number 3 (2004), 1727-1745.

Abstract

Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|G_n|, 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.

Primary Subjects: 05C80, 60K35

Keywords: Percolation; random graph; expander; giant component

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1089808409
Digital Object Identifier: doi:10.1214/009117904000000414
Mathematical Reviews number (MathSciNet): MR2073175
Zentralblatt MATH identifier: 1046.05071

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