On the truncated anisotropic long-range percolation on \(\mathbb Z^2\)
type: Item
instance of: scholarly article
title: On the truncated anisotropic long-range percolation on \(\mathbb Z^2\)
zbMATH Open document ID: 0963.60097
DOI: 10.1016/S0304-4149(98)00107-0
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: The following infinite-range bond percolation process on \(\mathbb Z^2\) is studied. Each \(x\in \mathbb Z^2\) is connected to both nearest neighbours in the vertical direction with probability \(\varepsilon >0\); each \(x\in \mathbb Z^2\) is connected to each vertex \(x{+\atop -}(i,0)\) with probability \(p_i\geq 0\), \(i\geq 1\); all different connections are independent. Given \(\varepsilon >0\) and let the truncated process \(K\) be defined from the original one by replacing \(p_i\), \(i> K\), by zero. The main result says that, if \(p_i\geq \frac {1}{i\ln i}\) for \(i\) sufficiently large, then for every \(\varepsilon >0\) there is a \(K\) such that the probability of the open cluster of the origin being infinite is positive for the corresponding truncated process. The method of proof is a reduction to anisotropic nearest neighbour Bernoulli percolation process with \(p_V+p_H>1\), where \(p_V\) corresponds to the vertical and \(p_H\) to the horizontal connections. The reduction uses some renormalization procedure in the horizontal direction. The study is motivated by the open problem for infinite-range percolation processes in \(\mathbb Z^d\) asking if the condition \(\sum _{y\in \mathbb Z^d} p_{\langle x,y\rangle}=\infty \) for every \(x\in \mathbb Z^d\) is sufficient for the existence of infinite open cluster of the origin for some truncated processes.
reviewed by: Petr Holický
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 1554319
zbMATH Keywords: long-range percolation
zbMATH Keywords: truncated process
zbMATH Keywords: renormalization
author: Vladas Sidoravicius
author: Donatas Surgailis
author: Maria Eulalia Vares
MaRDI profile type: Publication
cites work: Sharpness of the phase transition in percolation models
cites work: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation
cites work: Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models
cites work: Oriented percolation in two dimensions
cites work: On the connectedness of a random graph
cites work: Q3672870
cites work: On the continuity of the critical value for long range percolation in the exponential case
cites work: Q3792027
cites work: One dimensional \(1/| j-i| ^ s\) percolation models: The existence of a transition for s\(\leq 2\)
description: scientific article; zbMATH DE number 1554319