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An improved lower bound for the multidimensional dimer problem

1968; Cambridge University Press; Volume: 64; Issue: 2 Linguagem: Inglês

10.1017/s030500410004305x

1469-8064

J. M. Hammersley,

History and advancements in chemistry

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 a 1 , a 2 , …, a d , where n = a 1 a 2 … a d , and write a = ( a 1 , a 2 , …, a d ). Let f a denote the number of ways of dissecting this brick into ½ n dimers. On the basis of physical and heuristic arguments chemists have known for many years that f a increases more or less exponentially with n ; and recently a rigorous proof (1) of this fact has been given in the following form: if a i → ∞ for all i = 1, 2, …, d , then n −1 log f a tends to a finite limit, which we denote by λ d . The principal outstanding problem for chemists is to determine the numerical value of λ 3 , or failing an exact determination to estimate λ 3 or to find upper and lower bounds for it.