A new solution for the two dimensional dimer problem

TL;DR

This paper introduces a novel method based on the Cauchy-Binet theorem for solving the 2D dimer problem, providing explicit formulas with fewer factors and applications to square and hexagonal lattices.

Contribution

A new solution method for the 2D dimer problem using the Cauchy-Binet theorem, simplifying formulas and extending to hexagonal lattices.

Findings

Explicit product formulas for square lattice dimer counts.

Simpler formulas with fewer factors compared to classical methods.

Application to counting periodic stepped surfaces in hexagonal lattices.

Abstract

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph -- due independently to Kasteleyn and to Temperley and Fisher -- consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method for solving the two dimensional dimer problem, which relies on the Cauchy-Binet theorem. It only involves facts that were known in the mid 1930's when the dimer problem was phrased, so it could have been discovered while the dimer problem was still open. We provide explicit product formulas for both the square and the hexagonal lattice. One advantage of our formula for the square lattice compared to the original formula of Kasteleyn, Temperley and Fisher is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section, which can be regarded as a counterpart of MacMahon's boxed plane partition theorem.