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Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model. (English) Zbl 1318.82012

Summary: 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 R. S. Ellis and K. Wang [Stochastic Processes Appl. 35, No. 1, 59–79 (1990; Zbl 0705.60027)] 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 \(0<s<q\) classes and identifying the spins within these classes. In extension of earlier work [O. Häggström and C. Külske, Markov Process. Relat. Fields 10, No. 3, 477–506 (2004; Zbl 1210.82023)] 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 one or equal to three).
We also provide a dynamical interpretaion 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.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
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