The Annals of Probability

Percolation on finite graphs and isoperimetric inequalities

Noga Alon, Itai Benjamini, and Alan Stacey

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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.

Article information

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

Dates
First available in Project Euclid: 14 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1089808409

Digital Object Identifier
doi:10.1214/009117904000000414

Mathematical Reviews number (MathSciNet)
MR2073175

Zentralblatt MATH identifier
1046.05071

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

Keywords
Percolation random graph expander giant component

Citation

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. https://projecteuclid.org/euclid.aop/1089808409


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