Is the fuzzy Potts model Gibbsian?
Let \(q\) be a positive integer and \(\beta \) be the inverse temperature, and let \(s\) and \(r_1, \dots, r_s \) be positive integers such that \(\sum_{i=1}^s r_i =q\). The fuzzy Potts model on a finite graph \( G=(V, E) \) with these parameters arises by taking the \(q\)-state Potts model on \( G \) and then identifying \( r_1\) of the Potts states with a single fuzzy spin value 1, \(r_2 \) of the states with fuzzy spin value 2, and so on. This definition is a slight generalization, and a unification, of definitions given by \textit{C. Maes} and \textit{K. Vande Velde} [J. Phys. A, Math. Gen. 28, 4261--4271 (1995; Zbl 0868.60081)] (who considered the case \(r_1 =\cdots= r_s\)), and the author [Ann. Appl. Probab. 9, 1149--1159 (1999; Zbl 0957.60099)] (who considered the case \(s=2\)). The case where \(s=2\) and \(r_1 =1 \) has also been studied by \textit{L. Chayes} [Stochastic Processes Appl. 65, 209--216 (1996; Zbl 0889.60096)]. The induced probability measure \(\mu_{q, \beta}(r_1, \dots, r_s )^G \) is called the fuzzy Potts measure with parameters \(q, \beta \) and \((r_1, \dots, r_s)\). The main result of this paper is the following theorem. Consider the fuzzy Potts measure \(\mu_{q, \beta}(r_1, \dots,r_s )^{Z^d, i}\) with \(d \geq 2, q \geq 3\), fixed \(i \in \{0, \dots, q \}\) and \((r_1, \dots,r_s )\) satisfying \(1<s<q\), (i) For \(\beta < \beta_c (d, r_1)\), the measure \(\mu_{q, \beta} (r_1, \dots,r_s )^{Z^d, i}\) is a Gibbsian measure. (ii) In contrast, for \(\beta>\frac{1}{2}\log((1+(r_1 -1)p_c (d))/(1-p_c (d)))\), \(\mu_{q, \beta}(r_1, \dots,r_s )^{Z^d, i}\) is not a Gibbsian measure. Here \(p_c(d) \) denotes the critical value for bond percolation on \({\mathbb{Z}}^d\).
scientific article; zbMATH DE number 1975015