Gibbsianness and non-Gibbsianness in divide and color models

Details Gibbsianness and non-Gibbsianness in divide and color models Gibbsianness and non-Gibbsianness in divide and color models

2008-12-12

arxiv.org/abs/0812.2399v4

Annals of Probability 2010, Vol. 38, No. 4, 1609-1638

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP518

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 [$\operatorname {DaC}(q)$] 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 $\{1,2,\...,s\}$ 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 $q\in \{2,3,\...\}$, $a_1=a_2=\...=a_s=1/q$, when the $\operatorname {DaC}(q)$ model coincides with the $q$-state Potts model.

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