Gibbsianness and non-Gibbsianness in divide and color models

TL;DR

This paper investigates the Gibbsian properties of the divide and color model on or parameters where the FK measure is unique, showing it is Gibbs for small parameters and non-Gibbs for large ones, with special cases matching the Potts model.

Contribution

It characterizes when the divide and color model is Gibbs or non-Gibbs, extending understanding of phase transitions in these models.

Findings

Gibbs measure for small p

Non-Gibbs measure for large p

Special case matches Potts model

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 [$\operatorname {DaC}(q)$] 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.