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arXiv · 0708.1641

Approximating the monomer-dimer constants through matrix permanent

Yan Huo·Heng Liang·Si-Qi Liu·Fengshan Bai
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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\pm0.0002$, where the exact value is $h_2=0.662798972834$. For three-dimensional lattice with periodic condition, our numerical result is $ 0.7847\pm0.0014$, {which agrees with the best known bound $0.7653 \leq h_3 \leq 0.7862$.}

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Yan Huo, Heng Liang, Si-Qi Liu, Fengshan Bai. 2007-08-15. Approximating the monomer-dimer constants through matrix permanent. https://doi.org/10.1103/physreve.77.016706

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