Packing dimers on $(2p + 1) \times (2q + 1) $ lattices

TL;DR

This paper investigates the number of dimer packings on odd-by-odd lattices, revealing number-theoretical properties and a unique logarithmic correction in free energy, suggesting potential solvability of the monomer-dimer problem.

Contribution

It uncovers number-theoretical properties of dimer configurations on odd lattices and identifies a logarithmic term in free energy corrections, advancing understanding of the monomer-dimer problem.

Findings

Number-theoretical properties of dimer counts on odd lattices

Existence of a logarithmic correction in free energy

Potential implications for solving the monomer-dimer problem

Abstract

We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n \ge 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable.