Gibbsianness and non-Gibbsianness in divide and color models

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