Gibbsianness and non-Gibbsianness in divide and color models
András Bálint
Source: Ann. Probab. Volume 38, Number 4
(2010), 1609-1638.
Abstract
For parameters p ∈ [0, 1] and q > 0 such that the Fortuin–Kasteleyn (FK) random-cluster measure Φp,qℤd 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,qℤd. 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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Keywords: Divide and color models; Gibbs measures; non-Gibbsianness; quasilocality; random-cluster measures
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1278593962
Digital Object Identifier: doi:10.1214/09-AOP518
Mathematical Reviews number (MathSciNet): MR2663639
Zentralblatt MATH identifier: 1200.60084
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