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 $\mathbb{Z}^d$ withparameters $p$ 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 bondconfiguration with distribution $\Phi_{p,q}^{\mathbb{Z}^d}$. Then, to each (FK)cluster (i.e., to every vertex in the FK cluster), independently for differentFK clusters, we assign a spin value from the set $\{1,2,\...,s\}$ in such a waythat spin $i$ has probability $a_i$. In this paper, we prove that the resultingmeasure on spin configurations is a Gibbs measure for small values of $p$ andis 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 $\operatorname {DaC}(q)$ modelcoincides with the $q$-state Potts model.
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机译：对于参数$ p \ in [0,1] $和$ q> 0 $，以使Fortuin-Kasteleyn（FK）随机群集测量$ \ Phi_ {p，q} ^ {\ mathbb {Z} ^ d}参数$ p $和$ q $的$ \ mathbb {Z} ^ d $的$是唯一的，$ \ mathbb {Z}上的$ q $分度和颜色[$ \ operatorname {DaC}（q）$]模型^ d $定义如下。首先，我们绘制一个债券配置，其分布为$ \ Phi_ {p，q} ^ {\ mathbb {Z} ^ d} $。然后，对于每个（FK）集群（即，对于FK集群中的每个顶点），独立地针对不同的FK集群，我们从集合中的$ \ {1,2，\ ...，s \} $中分配旋转值这样旋转$ i $的概率就有$ a_i $。在本文中，我们证明了自旋配置的结果度量是对$ p $小值的Gibbs度量，而不是对于大$ p $的Gibbs度量，除非在特殊情况下$ q \ in \ {2,3， \ ... \} $，$ a_1 = a_2 = \ ... = a_s = 1 / q $，当$ \ operatorname {DaC}（q）$模型与$ q $状态的Potts模型同时出现时。
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