Periodic Gibbs measures for the soft-core Widom–Rowlinson model and their extremality

Source abstract

We consider both hard-core and soft-core Widom–Rowlinson models with spin values $(-1, 0, 1)$ on a Cayley tree of order $k \geq 2$, focusing on the Gibbs measures of these models. The models depend on three parameters: the order $k$ of the tree, the interaction strength $\theta$ (describing $J < 0$ and $J > 0$ cases), and the particle intensity $\lambda$. The hard-core Widom–Rowlinson model corresponds to the case $\theta = 0$. For $k = 2$ and $J > 0$, we prove the existence of two and four non-translation-invariant (non-TI) periodic Gibbs measures within the corresponding regions of the parameters $\theta$ and $\lambda$. Since the explicit forms of the four solutions to the system of functional equations are obtained, the conditions for the extremality and non-extremality of these periodic Gibbs measures are established. In the case $k = 3$ and $J > 0$, under the condition $\theta > 2$, explicit expressions for two critical values $\lambda_{c,t,i}(\theta)$ ($i = 1, 2$) are derived. Using these critical values, we show that there exist exactly two or exactly four non-translation-invariant periodic Gibbs measures in the corresponding regions of the parameter space.