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We study the natural linear operators associated to divide and color (DC)models. The degree of nonuniqueness of the random partition yielding a DC modelis directly related to the dimension of the kernel of these linear operators.We determine exactly the dimension of these kernels as well as analyze apermutation-invariant version. We also obtain properties of the solution setfor certain parameter values which will be important in (1) showing that largethreshold discrete Gaussian free fields are DC models and in (2) analyzing whenthe Ising model with a positive external field is a DC model, both in futurework. However, even here, we give an application to the Ising model on atriangle.

An analysis of the induced linear operators associated to divide and color models