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

2007/11/07 by Paul Federbush, Federbush, Paul
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0711.1092

openalex publication_date 2007/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)/d² +... with the first few terms given explicitly.

Citations

• Tilings With Very Elastic Tiles

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