Some results on the phase structure of the two-dimensional Widom-Rowlinson model
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 \(| h| >h_c\) then \(| \mathcal G(\lambda,h)| =1.\) If \(| \mathcal G(\lambda,0)| >1\) then \(h_c(\lambda)= 0\). The second result concerns the symmetric case (h=0) and is a partial description of the structure of the set of equilibrium states. Let \(p_c\) be the critical probability of Bernoulli site percolation. {\textbf{Theorem 2}} Consider \(\lambda>\frac {8p_c}{1-p_c}\) and let \(\mu^+_\lambda\) and \(\mu^-_\lambda\) be the limiting Gibbs measures with plus and minus boundary conditions. Then: (1) The limiting Gibbs measure with free boundary condition is equal to \(\frac {\mu^+_\lambda+\mu^-_\lambda}{2}\). (2) If \(\mu\in\mathcal G(\lambda,0)\) is either horizontally periodic or vertical periodic then \[ \mu=\alpha\mu^+_\lambda+(1-\alpha)\mu^-_\lambda \] with some \(\alpha\in[0,1]\).
scientific article; zbMATH DE number 2111455