Dimer problem for some three dimensional lattice graphs

摘要

Dimer problem for three dimensional lattice is an unsolved problem in statistical mechanics and solid-state chemistry. In this paper, we obtain asymptotical expressions of the number of close-packed dimers (perfect matchings) for two types of three dimensional lattice graphs. Let M(G) denote the number of perfect matchings of G. Then log(M(K-2 X C-4 X P-n)) approximate to (-1.171. n(-1.1223) + 3.146)n, and log(M(K-2 X P-4 X P-n)) approximate to (-1.164. n(-1.196) + 2.804)n, where log() denotes the natural logarithm. Furthermore, we obtain a sufficient condition under which the lattices with multiple cylindrical and multiple toroidal boundary conditions have the same entropy. (C) 2015 Elsevier B.V. All rights reserved.