Gibbsian and Non-Gibbsian Properties of the Generalized Mean-Field Fuzzy Potts-Model
2014, v.20, Issue 4, 601-632
We analyze the generalized mean-field q-state Potts model which is obtained by replacing the usual quadratic interaction function in the mean-field Hamiltonian by a higher power z. We first prove a generalization of the known limit result for the empirical magnetization vector of Ellis and Wang [R.S. Ellis and K.W. Wang, Limit theorems for the empirical vector of the Curie - Weiss - Potts model, Stoch. Process. Appl., 1989, v. 35, 59-79] which shows that in the right parameter regime, the first-order phase-transition persists. Next we turn to the corresponding generalized fuzzy Potts model which is obtained by decomposing the set of the $q$ possible spin-values into $1 < s< q$ classes and identifying the spins within these classes. In extension of earlier work [O. Haggstrom and C. Kuelske, Gibbs properties of the fuzzy Potts model on trees and in mean field, Markov Processes Relat. Fields, 2004, v. 10, N 3, 477-506] which treats the quadratic model we prove the following: The fuzzy Potts model with interaction exponent bigger than four (respectively bigger than two and smaller or equal four) is non-Gibbs if and only if its inverse temperature $\beta$ satisfies $\beta \geq \beta_c(r_*,z)$ where $\beta_c(r_*,z)$ is the critical inverse temperature of the corresponding Potts model and $r_*$ is the size of the smallest class which is greater than or equal to two (respectively greater than or equal to three). We also provide a dynamical interpretation considering sequences of fuzzy Potts models which are obtained by increasingly collapsing classes at finitely many times $t$ and discuss the possibility of a multiple in- and out of Gibbsianness, depending on the collapsing scheme.
Keywords: Potts model,Fuzzy Potts model,Ellis - Wang Theorem,Gibbsian measures,non-Gibbsian measures,mean-field measures