Kesten's incipient infinite cluster and quasi-multiplicativity of\n crossing probabilities

2016/03/22 by Deepan Basu, Basu, Deepan, Artëm Sapozhnikov +1 · 1 citation
Mathematics · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1603.06884

Abstract

In this paper we consider Bernoulli percolation on an infinite connected\nbounded degrees graph G. Assuming the uniqueness of the infinite open cluster\nand a quasi-multiplicativity of crossing probabilities, we prove the existence\nof Kesten's incipient infinite cluster. We show that our assumptions are\nsatisfied if G is a slab mathbb Z²\× 0,\…,k ^d-2 (d\≥ 2,\nk\≥ 0). We also argue that the quasi-multiplicativity assumption is\nfulfilled for G= mathbb Z^d if and only if d<6.\n

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