Approximating the monomer-dimer constants through matrix permanent

TL;DR

This paper introduces a novel matrix permanent formulation for the monomer-dimer model's partition function, enabling approximation of monomer-dimer constants in 2D and 3D lattices using importance sampling.

Contribution

It proposes a new matrix permanent-based method for approximating monomer-dimer constants, transforming matchings into perfect matchings on extended graphs.

Findings

Achieved a 2D lattice constant of 0.6627±0.0002, close to the exact value.

Estimated the 3D lattice constant as 0.7847±0.0014, aligning with known bounds.

Demonstrated the effectiveness of importance sampling for permanent computation.

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 $0.7653 \leq h_3 \leq 0.7862$.}