Bond percolation in frustrated systems
type: Item
instance of: scholarly article
title: Bond percolation in frustrated systems
zbMATH Open document ID: 0968.60092
author: N. Delaunay
published in: The Annals of Probability
publication date: 2001-09-02T00:00:00Z
review text: As is well known, the presence of phase transition in classical ferromagnetic (Ising, Potts) systems at low temperatures is equivalent to percolation in the related random cluster model. For non-ferromagnetic systems, like the famous Edwards-Anderson spin glass model on \({\mathbb Z}^d\), the presence of negative couplings introduces long range correlations between bonds in the related random cluster model. Such dependence becomes decisive in the zero temperature limit since it forbids any bond configuration that contains frustrated circuits (a circuit is called frustrated if the product of all its coupling constants is negative). It thus becomes natural and important to study percolation in random cluster models subject to ``non-frustration'' property. The simplest situation of this type occurs as follows: to any bond \(b\) of \({\mathbb Z}^d\) assign a value \(J_b\in\{-1,+1\}\) and consider the usual percolation model at density \(p\) of occupied bonds conditioned to avoid full occupation of frustrated circuits. Using a variant of Peierls' argument, the authors show that if \(p>8/9\), then uniformly in all configurations \(\{J_b\}\) the percolation of occupied bonds in such a system takes place with probability one. A similar result is obtained for the (conditioned on ``non-frustration'') FK random cluster measure on~\({\mathbb Z}^d\). To stress the fact that the percolation/non-percolation transition in systems with frustration is not necessarily sharp, the authors show that for a particular graph and a particular choice of the values assumed by \(J\), it is possible to identify four regions of the interval \((0,1)\) such that if \(p\) is in these regions, percolation occurs for none, almost none, almost all or all of the \(J\)'s, respectively. Nontrivial estimates for the lengths of these intervals are given for the triangular lattice and \(J\in\{-t,-1,+1,+t\}\) with~\(t>0\) .
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 1496107
zbMATH Keywords: bond percolation
zbMATH Keywords: FK measure
zbMATH Keywords: frustration
zbMATH Keywords: Peierls' argument
zbMATH Keywords: spin glasses
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677548
OpenAlex ID: W4230816053
DOI: 10.1214/AOP/1022677548
description: scientific article; zbMATH DE number 1496107