Some results on the phase structure of the two-dimensional Widom-Rowlinson model
type: Item
instance of: scholarly article
title: Some results on the phase structure of the two-dimensional Widom-Rowlinson model
zbMATH Open document ID: 1088.82006
author: Y. Higuchi
author: M. Takei
published in: Osaka Journal of Mathematics
publication date: 2004-10-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1153493511
review text: The authors study the 2D-lattice Widom-Rowlinson model. More precisely they consider a three value spin model with configuration space \(\Omega=\{-1,0,+1\}^{\mathbb Z^2}\) endowed with the product topology. The interaction is given via the notion for a configuration to be feasible. A configuration \(\omega\in\Omega\) is feasible in the region \(\Lambda\subset\mathbb Z^2\) if \(\omega(x)\omega(y)\neq -1\) for all \(x,y\in\Lambda\) such that \(| x-y| \equiv| x_1-y_1| +| x_2-y_2| =1\). Given a finite subset \(\Lambda\) of \(\mathbb Z^2\) and a feasible boundary condition \(\omega\in\Omega\), the finite volume Gibbs distribution \(\mu_{\Lambda,\lambda,h}^\omega(\sigma)\) is defined by \[ \mu_{\Lambda,\lambda,h}^\omega(\sigma)=\frac {1}{Z_{\Lambda,\lambda,h}^\omega} \text{\textbf{ {1}}}_{\{\sigma\star\omega:\text{ feasible}\}} \prod_{x\in\Lambda}\lambda^{\sigma(x)^2}e^{h\sigma(x)}. \] Here \(\lambda >0\) is called activity and \(h\in\mathbb R\) is similar to the usual external field in the Ising model. \(Z_{\Lambda,\lambda,h}^\omega\) is the partition function. \({\text{\textbf{1}}_A}\) is the characteristic function of the set \(A\). The configuration \(\sigma\star\omega\in\Omega\) is defined by: \[ \sigma\star\omega(x)=\begin{cases} \sigma(x)&\text{ if }x\in\Lambda\\ \omega(x) &\text{ if }x\in\Lambda^c.\end{cases} \] A probability measure \(\mu\) on \((\Omega,\mathcal F)\) which satisfies the DLR (Dobrushin-Lanford-Ruelle) equation (\(\Lambda \) bounded subset of \(Z^2\)) \[ \mu(\cdot\;| \mathcal F^{\Lambda^c})(\omega)=\mu_{\Lambda,\lambda,h}^\omega(\cdot)\;\;\;\mu-a.a.\;\omega \] it is said to be the Gibbs measure with parameter \((\lambda,h)\). Let \(\mathcal G(\lambda,h)\) denote the set of all Gibbs measures with parameters \((\lambda,h)\). The results concern the problem of absence and existence of phase transitions. The first result is a uniqueness one. {\textbf{Theorem 1}} For each \(\lambda >0\), there exists \(h_c=h_c(\lambda)\geq 0\) such that if \(| h| >h_c\…
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 2111455
zbMATH Keywords: statistical mechanics
zbMATH Keywords: Gibbs state
zbMATH Keywords: phase transition
zbMATH Keywords: percolation
reviewed by: A. Pellegrinotti
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2111455