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.

11 pages, new appendix with comparison to known rigorous results

References in corpus (6)

• Validations of the Asymptotic Matching Conjectures
• Dimer lambda_3 = .453 +/- .001 and Some Other Very Intelligent Guesses
• Dimer lambda_d Expansion Computer Computations
• Dimer lambda_d Expansion, Dimensional Dependence of J_n Kernels
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Cited by in corpus (6)

• Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices
• An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
• Convergence of the Formal Expansion for lambda_d(p) of the Monomer-Dimer Problem for Small p
• Power series expansions for the planar monomer-dimer problem
• Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems
• Matchings in vertex-transitive bipartite graphs