Planar Graphical Models which are Easy

TL;DR

This paper introduces a class of planar graphical models for binary variables that are computationally efficient to analyze, leveraging Gaussian Grassmann models and Pfaffian evaluations, thus enabling polynomial-time partition function calculations.

Contribution

It establishes a connection between binary variable models on planar graphs and Gaussian Grassmann models, providing a polynomial-time method to compute their partition functions.

Findings

Partition function calculation reduces to Pfaffian evaluation.

Models are equivalent to Gaussian Grassmann models on the same graph.

Method applies to a rich family of binary variable models on planar graphs.

Abstract

We describe a rich family of binary variables statistical mechanics models on a given planar graph which are equivalent to Gaussian Grassmann Graphical models (free fermions) defined on the same graph. Calculation of the partition function (weighted counting) for such a model is easy (of polynomial complexity) as reducible to evaluation of a Pfaffian of a matrix of size equal to twice the number of edges in the graph. In particular, this approach touches upon Holographic Algorithms of Valiant and utilizes the Gauge Transformations discussed in our previous works.