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July 2010 Gibbsianness and non-Gibbsianness in divide and color models
András Bálint
Ann. Probab. 38(4): 1609-1638 (July 2010). DOI: 10.1214/09-AOP518


For parameters p ∈ [0, 1] and q > 0 such that the Fortuin–Kasteleyn (FK) random-cluster measure Φp,qd for ℤd with parameters p and q is unique, the q-divide and color [DaC(q)] model on ℤd is defined as follows. First, we draw a bond configuration with distribution Φp,qd. Then, to each (FK) cluster (i.e., to every vertex in the FK cluster), independently for different FK clusters, we assign a spin value from the set {1, 2, …, s} in such a way that spin i has probability ai.

In this paper, we prove that the resulting measure on spin configurations is a Gibbs measure for small values of p and is not a Gibbs measure for large p, except in the special case of q ∈ {2, 3, …}, a1 = a2 = ⋯ = as = 1/q, when the DaC(q) model coincides with the q-state Potts model.


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András Bálint. "Gibbsianness and non-Gibbsianness in divide and color models." Ann. Probab. 38 (4) 1609 - 1638, July 2010.


Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1200.60084
MathSciNet: MR2663639
Digital Object Identifier: 10.1214/09-AOP518

Primary: 60K35 , 82B20 , 82B43

Keywords: Divide and color models , Gibbs measures , Non-Gibbsianness , quasilocality , random-cluster measures

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 4 • July 2010
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