Paul Federbush
Hidden Structure in Tilings, Conjectured Asymptotic Expansion
for lambda_d in Multidimensional Dimer Problem
(14K, LaTeX)
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%.