Dense dimer packings in the 3D cubic lattice

Abstract

The dimer model, originally introduced by Fowler and Rushbrooke to describe diatomic adsorption on crystal surfaces, has since evolved into a central object in statistical mechanics, with deep connections to graph theory, combinatorics, and computational complexity. While exact enumeration of dimer packings in two-dimensional lattices was achieved by Kasteleyn, Temperley, and Fisher, in a closed-form formula, the three-dimensional analog on the simple cubic lattice remains far less understood due to the loss of planarity and the exponential growth of configuration space. In this paper, we establish a recursive, transfer-matrix-based framework for the exact enumeration of dense dimer packings in finite regions of the cubic lattice. Specifically, we define and analyze a class of nested matrices equipped with a novel tensor product structure that captures the combinatorial dynamics of dimer arrangements in three dimensions. The resulting recurrence relations yield exact enumeration formulas for parallelepiped domains of arbitrary size, thus providing a rigorous combinatorial foundation for studying three-dimensional lattice tilings.