Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model

Abstract

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 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 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 is larger or equal than the critical inverse temperature of the corresponding Potts model with number of spin-labels equal to the size of the smallest class which is greater or equal than two (respectively greater or equal than three.) We also provide a dynamical interpretation considering sequences of fuzzy Potts models which are obtained by increasingly collapsing classes at finitely many times and discuss the possibility of a multiple in- and out of Gibbsianness, depending on the collapsing scheme.

26 pages, 6 figures

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• Rapid Mixing of Glauber Dynamics of Gibbs Ensembles via Aggregate Path Coupling and Large Deviations Methods
• The aggregate path coupling method for the Potts model on bipartite graph
• Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3
• Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice