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

TL;DR

This paper proves the existence of Kesten's incipient infinite cluster in certain graphs under assumptions of uniqueness and quasi-multiplicativity, which are verified for specific lattice structures and dimensions.

Contribution

It establishes conditions for the existence of the incipient infinite cluster and verifies these conditions for slabs and discusses their validity in high dimensions.

Findings

Quasi-multiplicativity holds for $\\mathbb{Z}^2$ and slabs.

Existence of incipient infinite cluster is proven under certain assumptions.

Quasi-multiplicativity fails for $d\geq 6$ in $\mathbb{Z}^d$.

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 $d<6$.