Dependent percolation in two dimensions
type: Item
instance of: scholarly article
title: Dependent percolation in two dimensions
zbMATH Open document ID: 0965.60096
author: Béla Bollobás
author: Paul Balister
author: Alan M. Stacey
publication date: 2001-03-20T00:00:00Z
review text: Let \(k\) be a natural number and define an oriented site percolation on \(Z^2\) as follows. Let \(x_i\), \(y_j\) be independent random variables with values uniformly distributed in \(\{1,\ldots,k\}\). Declare a site \((i,j)\in Z^2\) closed if \(x_i=y_j\), and open otherwise. P. Winkler conjectured that if \(k\geq 4\), then with positive probability there is an infinite path \(P=(i_0,j_0)(i_1,j_1)\ldots\) such that \(0=i_0\leq i_1\leq \ldots\), \(0=j_0\leq j_1\leq \ldots\), and each site \((i_n,j_n)\) is open. This conjecture is still open. In this paper the authors prove the weaker result that if \(k\geq 4\), then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1525164
zbMATH Keywords: dependent percolation
zbMATH Keywords: uniformly distributed random variables
reviewed by: N. G. Gamkrelidze
MaRDI profile type: Publication
DOI: 10.1007/S004400000060
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1525164