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Percolation on finite graphs and isoperimetric inequalities

Noga Alon·Itai Benjamini·Alan Stacey
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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.

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Noga Alon, Itai Benjamini, Alan Stacey. 2005-03-30. Percolation on finite graphs and isoperimetric inequalities. https://doi.org/10.1214/009117904000000414

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