07-288 Paul Federbush

Hidden Structure in Tilings, Conjectured Asymptotic Expansion for lambda_d in Multidimensional Dimer Problem (14K, LaTeX) Nov 21, 07

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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 the next two terms in the expansion to be given as in lambda_d ~ (1/2)ln(2d) -1/2 +(1/8)/d + (7/48)/d^2 Although this is an expansion designed for large d, even at d=2 it is remarkably accurate, in error by about .3%.

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