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Percolation on finite graphs and isoperimetric inequalities | | Noga Alon
; Itai Benjamini
; Alan Stacey
; | | Date: |
12 Jul 2002 | | Journal: | Annals of Probability 2004, Vol. 32, No. 3, 1727-1745 DOI: 10.1214/009117904000000414 | | Subject: | Probability; Combinatorics MSC-class: 05C80, 60K35 (Primary) | math.PR math.CO | | 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, Komlos and Szemeredi [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. | | Source: | arXiv, math.PR/0207112 | | Services: | Forum | Review | PDF | Favorites | |
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