Combinatorics of the Dimer Model on a Strip

TL;DR

This paper derives a closed-form formula for the partition function of the dimer model on a 2 by n strip of squares or hexagons, using combinatorial and recursive methods, with implications for counting perfect matchings.

Contribution

It provides a novel closed-form expression for the dimer partition function on a strip, connecting combinatorial, statistical mechanics, and graph theory approaches.

Findings

Closed-form formula for the partition function

Derivation via Potts model-like description and recursion

Connection to minimal feedback arc set problem

Abstract

In this note, we give a closed formula for the partition function of the dimer model living on a (2 x n) strip of squares or hexagons on the torus for arbitrary even n. The result is derived in two ways, by using a Potts model like description for the dimers, and via a recursion relation that was obtained from a map to a 1D monomer-dimer system. The problem of finding the number of perfect matchings can also be translated to the problem of finding a minmal feedback arc set on the dual graph.