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Approximating the number of monomer-dimer coverings of a lattice

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Abstract

We study the problem of counting the number of coverings of ad-dimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn, and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattices with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in constrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do not rely on any assumptions. The method generalizes to counting coverings of any finite vertex-transitive graph, a class which includes most natural finite lattices with periodic boundary conditions.

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Authors and Affiliations

  1. CNRS, Ecole Normale Supérieure de Lyon, France

    Claire Kenyon

  2. Department of Computer Science, Princeton University, Princeton, New Jersey

    Dana Randall

  3. Computer Science Division, University of California at Berkeley, Berkeley, California

    Alistair Sinclair

Authors
  1. Claire Kenyon
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  2. Dana Randall
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  3. Alistair Sinclair
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Kenyon, C., Randall, D. & Sinclair, A. Approximating the number of monomer-dimer coverings of a lattice. J Stat Phys 83, 637–659 (1996). https://doi.org/10.1007/BF02183743

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