An Improved Upper Bound For The Three Dimensional Dimer Problem

. Let n be even and denote by f(n) the number of domino tilings of a cube of side n. The three dimensional dimer problem is to determine the limit ` 3 := limn!1 (log f(n))=n 3 (which is known to exist). The best previously known upper bound was found by Minc and is 1=12 log 6! = 0:54827:::. In this paper we improve this bound to 0.463107. 1. Introduction The dimer problem, which in dimension three is one of the classical unsolved problems in solid-state chemistry, is the following. Define an brick to be a d-dimensional (d 2) rectangular parallelepiped with sides of integer lengths; we will always assume its volume to be even. A brick of volume 2 is called a dimer. The problem is to determine the number f(a 1 ; : : : ; a d ) of dimer tilings of the brick with sides of lengths a 1 ; : : : ; a d . It is remarkable that for d = 2 this number can be expressed in closed form, as was shown independently by Fisher, Kasteleyn and Temperley (see [14] and [6]). In contrast, for d 3 no such..