On the uniqueness of the infinite occupied cluster in dependent two- dimensional site percolation
type: Item
instance of: scholarly article
title: On the uniqueness of the infinite occupied cluster in dependent two- dimensional site percolation
zbMATH Open document ID: 0658.60133
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: This paper considers dependent site percolation on the two-dimensional square lattice. The underlying probability measure is assumed to be invariant and ergodic under either horizontal or vertical translation separately, and invariant under axis reflections. If the FKG condition is satisfied and if percolation occurs, then the infinite occupied cluster is unique with probability 1. The proof uses Harris' technique for showing uniqueness of the infinite percolation cluster, i.e., demonstrating the existence of circuits of occupied sites surrounding arbitrarily large boxes. A multiple ergodic lemma similar to a result of Furstenburg provides key inequalities for the probabilities that certain chains of occupied sites avoid the boxes. Symmetries and the FKG inequality then build the chains into closed circuits around the boxes. It is not known whether the result generalizes to higher dimensions than two, nor whether the hypotheses of the theorem can be weakened.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4076275
zbMATH Keywords: site percolation
zbMATH Keywords: FKG condition
zbMATH Keywords: infinite percolation cluster
zbMATH Keywords: multiple ergodic lemma
zbMATH Keywords: FKG inequality
author: Alberto Gandolfi
author: Michael Keane
author: Lucio Russo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991681
OpenAlex ID: W2010191561
DOI: 10.1214/AOP/1176991681
description: scientific article; zbMATH DE number 4076275