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An Analysis of the Induced Linear Operators Associated to Divide and Color Models

Author

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  • Malin P. Forsström

    (Chalmers University of Technology and Gothenburg University
    KTH Royal Institute of Technology)

  • Jeffrey E. Steif

    (Chalmers University of Technology and Gothenburg University)

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.

Suggested Citation

  • Malin P. Forsström & Jeffrey E. Steif, 2021. "An Analysis of the Induced Linear Operators Associated to Divide and Color Models," Journal of Theoretical Probability, Springer, vol. 34(2), pages 1043-1060, June.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:2:d:10.1007_s10959-020-01001-4
    DOI: 10.1007/s10959-020-01001-4
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    Keywords

    Divide and color models;

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