Gibbsianness and non-Gibbsianness in divide and color models

Abstract

For parameters p ∈ [0, 1] and q > 0 such that the Fortuin–Kasteleyn (FK) random-cluster measure Φ_p,q^ℤd for ℤ^d with parameters p and q is unique, the q-divide and color [DaC(q)] model on ℤ^d is defined as follows. First, we draw a bond configuration with distribution Φ_p,q^ℤd. 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 ∈ {2, 3, …}, a₁ = a₂ = ⋯ = a_s = 1/q, when the DaC(q) model coincides with the q-state Potts model.