Mappings for Marginal Probabilities with Applications to Models in Statistical Physics

TL;DR

This paper introduces a Fourier transform-based mapping between primal and dual normal factor graphs to efficiently estimate marginal probabilities in models like the Ising and Potts models, especially near phase transitions.

Contribution

It develops a novel local mapping technique using Fourier transforms to relate marginals in primal and dual graphs, improving estimation efficiency and accuracy in statistical physics models.

Findings

Mapping enables transforming marginals between primal and dual domains.

The method accurately captures phase transition points in the Ising model.

Numerical experiments show improved marginal probability estimates.

Abstract

We present local mappings that relate the marginal probabilities of a global probability mass function represented by its primal normal factor graph to the corresponding marginal probabilities in its dual normal factor graph. The mapping is based on the Fourier transform of the local factors of the models. Details of the mapping are provided for the Ising model, where it is proved that the local extrema of the fixed points are attained at the phase transition of the two-dimensional nearest-neighbor Ising model. The results are further extended to the Potts model, to the clock model, and to Gaussian Markov random fields. By employing the mapping, we can transform simultaneously all the estimated marginal probabilities from the dual domain to the primal domain (and vice versa), which is advantageous if estimating the marginals can be carried out more efficiently in the dual domain. An example of particular significance is the ferromagnetic Ising model in a positive external magnetic field. For this model, there exists a rapidly mixing Markov chain (called the subgraphs--world process) to generate configurations in the dual normal factor graph of the model. Our numerical experiments illustrate that the proposed procedure can provide more accurate estimates of marginal probabilities of a global probability mass function in various settings.