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

Abstract

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten's incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb Z^2\times\{0,\ldots,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption is fulfilled for $G=\mathbb Z^d$ if and only if $d<6$.