Gibbsianness and non-Gibbsianness in divide and color models

Abstract

For parameters $p\in [0,1]$ and $q>0$ such that the Fortuin--Kasteleyn (FK) random-cluster measure ${\Phi }_{p,q}^{{\mathbb{Z}}^d}$ for ${\mathbb{Z}}^d$ with parameters $p$ and $q$ is unique, the $q$-divide and color [$\mathrm{DaC}(q)$] model on ${\mathbb{Z}}^d$ is defined as follows. First, we draw a bond configuration with distribution ${\Phi }_{p,q}^{{\mathbb{Z}}^d}$. 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 $\{1,2,\dot{.}.,s\}$ in such a way that spin $i$ has probability $a_i$. 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 $q\in \{2,3,\dot{.}.\}$, $a_1=a_2=\dot{.}.=a_s=1/q$, when the $\mathrm{DaC}(q)$ model coincides with the $q$-state Potts model.

Published in at http://dx.doi.org/10.1214/09-AOP518 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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