Journal of Statistical Physics

, Volume 83, Issue 3–4, pp 637–659 | Cite as

Approximating the number of monomer-dimer coverings of a lattice

  • Claire Kenyon
  • Dana Randall
  • Alistair Sinclair
Articles

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.

Key Words

Monomer-dimer problem dimer coverings lattice statistics Monte Carlo methods relaxation time mixing time approximation algorithm Fisher-Kasteleyn-Temperley algorithm perfect matching monomer-dimer correlations vertex-transitive graphs 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Claire Kenyon
    • 1
  • Dana Randall
    • 2
  • Alistair Sinclair
    • 3
  1. 1.CNRSEcole Normale Supérieure de LyonFrance
  2. 2.Department of Computer SciencePrinceton UniversityPrinceton
  3. 3.Computer Science DivisionUniversity of California at BerkeleyBerkeley

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