## The Annals of Probability

### Percolation on finite graphs and isoperimetric inequalities

#### 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

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

Digital Object Identifier
doi:10.1214/009117904000000414

Mathematical Reviews number (MathSciNet)
MR2073175

Zentralblatt MATH identifier
1046.05071

#### 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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