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

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 on the lattice to completely cover it. It is known that the number of such coverings is roughly e*dV for some number *d. The first terms in the expansion of *d have been known for about thirty years *d, 12 ln(2d)- 12. Herein we present a mathematical argument for the next two terms in the expansion to be given as in *d, 12 ln(2d)- 12 + 18 1d + 748 1d2. Although this is an expansion designed for large d, even at d = 2 it is remarkably accurate, in error by about.3%.