The Annals of Probability

Percolation on finite graphs and isoperimetric inequalities

Noga Alon, Itai Benjamini, and Alan Stacey

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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.

Article information

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

First available in Project Euclid: 14 July 2004

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Percolation random graph expander giant component


Alon, Noga; Benjamini, Itai; Stacey, Alan. Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 (2004), no. 3, 1727--1745. doi:10.1214/009117904000000414.

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