Hidden Structure in Tilings, Conjectured Asymptotic Expansion for lambda_d in Multidimensional Dimer Problem

TL;DR

This paper proposes an asymptotic expansion for lambda_d in the multidimensional dimer problem, extending known approximations with explicit higher-order terms, based on a mathematical argument.

Contribution

It introduces a new asymptotic expansion for lambda_d in the multidimensional dimer problem, including explicit higher-order terms.

Findings

Derived an asymptotic expansion for lambda_d with explicit terms

Extended the known approximation from (1/2)ln(2d)-1/2 to include higher-order corrections

Provided a mathematical argument supporting the expansion

Abstract

The dimer problem arose in a thermodynamic study of diatomic molecules, and was abstracted into one of the most basic and natural problems in both statistical mechanics and combinatoric mathematics. Given a rectangular lattice of volume V in d dimensions, the dimer problem loosely speaking is to count the number of different ways dimers (dominoes) may be layed down on the lattice to completely cover it. It is known that the number of such coverings is roughly exp(lambda_d V) for some number lambda_d. The first terms in the expansion of lambda_d have been known for about thirty years lambda_d ~ (1/2)ln(2d)-1/2 Herein we present a mathematical argument for an asymptotic expansion lambda_d ~ (1/2)ln(2d) -1/2 +(1/8)/d + (5/96)/d^2 +... with the first few terms given explicitly.