Approximating the monomer-dimer constants through matrix permanent
arXiv:0708.1641 · doi:10.1103/PhysRevE.77.016706
Abstract
The monomer-dimer model is fundamental in statistical mechanics. However, it is $#P$-complete in computation, even for two dimensional problems. A formulation in matrix permanent for the partition function of the monomer-dimer model 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, we obtain , where the exact value is . For three-dimensional lattice with periodic condition, our numerical result is , {which agrees with the best known bound .}
6 pages, 2 figures