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

TL;DR

This paper investigates the linear operators linked to divide and color models, determining their kernels' dimensions and exploring properties relevant to Gaussian free fields and Ising models, advancing understanding of these probabilistic models.

Contribution

It precisely characterizes the kernels of the linear operators associated with divide and color models and analyzes their properties, including permutation-invariant versions.

Findings

Exact dimension of the kernels determined

Properties of solution sets for specific parameters analyzed

Application to Ising model on a triangle provided

Abstract

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