Open Access
Translator Disclaimer
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


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.


Download Citation

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


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

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

Primary: 05C80 , 60K35

Keywords: expander , Giant component , percolation , random graph

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • July 2004
Back to Top