Approximating the monomer-dimer constants through matrix permanent

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 $0.6627\pm 0.0002$, where the exact value is $h_2=0.662798972834$. For three-dimensional lattice with periodic condition, our numerical result is $0.7847\pm 0.0014$, {which agrees with the best known bound $0.7653\le h_3\le 0.7862$.}

6 pages, 2 figures

References in corpus (1)

• Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

Cited by in corpus (8)

• Sequential cavity method for computing free energy and surface pressure
• Yang-Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 <= d <= 7
• Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy
• Kosterlitz-Thouless transitions and phase diagrams of the interacting monomer-dimer model on a checkerboard lattice
• SUperman: Efficient Permanent Computation on GPUs
• State matrix recursion method and monomer--dimer problem
• Errata and Addenda to Mathematical Constants
• An analysis of a random algorithm for estimating all the matchings