Planar Graphical Models which are Easy

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.

💡 Research Summary

The paper “Planar Graphical Models which are Easy” introduces a broad class of binary‑variable statistical‑mechanics models defined on a planar graph and shows that each such model can be transformed exactly into a Gaussian Grassmann graphical model, i.e., a free‑fermion system, on the same graph. The central contribution is a constructive procedure that maps the original spin (Ising‑type) interaction into a purely quadratic form in Grassmann variables by means of local gauge transformations. Once the model is expressed as a quadratic Grassmann action, its partition function reduces to the Pfaffian of a skew‑symmetric matrix whose dimension is twice the number of edges. Because a Pfaffian can be evaluated in O(N³) time (with N = 2|E|) using standard Gaussian elimination, the weighted counting problem for the original binary model becomes polynomial‑time solvable.

The authors begin by recalling the definition of a Gaussian Grassmann graphical model: a set of anticommuting Grassmann variables ψₑ¹, ψₑ² attached to each edge e, with an action S = ½ ψᵀ A ψ, where A is a skew‑symmetric matrix. The partition function Z = ∫