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

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.