Computation of Terms in the Asymptotic Expansion of Dimer lambda_d for High Dimension

TL;DR

This paper derives an asymptotic expansion for the dimer problem's growth constant lambda_d in high dimensions, providing initial terms and discussing computational challenges for further terms.

Contribution

It introduces a mathematical approach for asymptotic expansion of lambda_d in inverse powers of dimension d, with computed initial terms and insights into computational limitations.

Findings

Derived asymptotic expansion for lambda_d in high dimensions

Computed initial terms of the series using computer calculations

Highlighted computational challenges for higher-order terms

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 laid down in the lattice (without overlapping) to completely cover it. Each dimer covers two neighboring vertices. It is known that the number of such coverings is roughly exp(lambda_d V) for some constant lambda_d as V goes to infinity. Herein we present a mathematical argument for an asymptotic expansion for lambda_d in inverse powers of d, and the results of computer computations for the first few terms in the series. As a glaring challenge, we conjecture no one will compute the next term in the series, due to the requisite computer time and storage demands.