An improved lower bound for the multidimensional dimer problem

Abstract

The dimer problem, which in the three-dimensional case is one of the classical unsolved problems of solid-state chemistry, can be formulated mathematically as follows. We define a brick to be a d-dimensional (d ≥ 2) rectangular parallelopiped with sides whose lengths are integers. An n-brick is a brick whose volume is n; and a dimer is a 2-brick. The problem is to determine the number of ways of dissecting an n-brick into dimers; and since this is only possible when n is even we confine attention hereafter to n-bricks with n even. Consider an n-brick with sides of length a1, a2, …, ad, where n = a1a2 … ad, and write a = (a1, a2, …, ad). Let fa denote the number of ways of dissecting this brick into ½n dimers. On the basis of physical and heuristic arguments chemists have known for many years that fa increases more or less exponentially with n; and recently a rigorous proof (1) of this fact has been given in the following form: if ai → ∞ for all i = 1, 2, …, d, then n−1 logfa tends to a finite limit, which we denote by λd. The principal outstanding problem for chemists is to determine the numerical value of λ3, or failing an exact determination to estimate λ3 or to find upper and lower bounds for it.