Computing monomer-dimer systems through matrix permanent.

The monomer-dimer model is fundamental in statistical mechanics. However, it is #P -complete in computation, even for two-dimensional problems. A formulation for the partition function of the monomer-dimer system is proposed in this paper by transforming the number of all matchings of a bipartite graph into the number of perfect matchings of an extended bipartite graph, which can be given by a matrix permanent. Sequential importance sampling algorithm is applied to compute the permanents. For two-dimensional lattice with periodic condition, the monomer-dimer constant is known as h_{2}=0.662798972834 . We obtain 0.6627+/-0.0002 for our approximation, which shows the robustness and the efficiency of the algorithm. For three-dimensional problem, our numerical result is 0.7847+/-0.0014 , which agrees with the best known bounds.