In Steif and Tykesson (J Prob 16:899–955, 2019), the authors introduced the so-called general divide and color models. One of the best-known examples of such a model is the Ising model with external field \( h = 0 \), which has a color representation given by the random cluster model. In this paper, we give necessary and sufficient conditions for this color representation to be unique. We also show that if one considers the Ising model on a complete graph, then for many \( h > 0 \), there is no color representation. This shows, in particular, that any generalization of the random cluster model which provides color representations of Ising models with external fields cannot, in general, be a generalized divide and color model. Furthermore, we show that there can be at most finitely many \( \beta > 0 \) at which the random cluster model can be continuously extended to a color representation for \( h \not = 0 \).

在 Steif 和 Tykesson (J Prob 16:899–955, 2019) 中，作者介绍了所谓的一般划分和颜色模型。这种模型最著名的例子之一是具有外部场\( h = 0 \)的 Ising 模型，它具有由随机聚类模型给出的颜色表示。在本文中，我们给出了这种颜色表示唯一的充要条件。我们还表明，如果在完整图上考虑 Ising 模型，那么对于许多\( h > 0 \)，没有颜色表示。这尤其表明，提供具有外部场的 Ising 模型的颜色表示的随机聚类模型的任何泛化通常不能是泛化的划分和颜色模型。此外，我们表明最多可以有有限多个\( \beta > 0 \)，在这些 \( \beta > 0 \)处，随机聚类模型可以连续扩展到\( h \not = 0 \)的颜色表示。