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Journal of Statistical Physics

, Volume 163, Issue 2, pp 211–238 | Cite as

A Pfaffian Formula for Monomer–Dimer Partition Functions

  • Alessandro Giuliani
  • Ian Jauslin
  • Elliott H. Lieb
Article

Abstract

We consider the monomer–dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the vertices and edges of the graph.

Keywords

Monomer–dimer problem Pfaffian formula Boundary monomers 

Notes

Acknowledgments

Thanks go to Jan Philip Solovej and Lukas Schimmer for devoting their time to some preliminary calculations that helped put us on the right track. We also would like to thank Tom Spencer and Joel Lebowitz for their hospitality at the IAS in Princeton and their continued interest in this problem. In addition, we thank Michael Aizenman and Hugo Duminil-Copin for discussing their work in progress on the random current representation for planar lattice models with us. We thank Jacques Perk and Fa Yueh Wu for very useful historical comments. We gratefully acknowledge financial support from the A*MIDEX project ANR-11-IDEX-0001-02 (A.G.), from the PRIN National Grant Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions (A.G. and I.J.), and NSF grant PHY-1265118 (E.H.L.).

References

  1. 1.
    Aizenman, M.: Geometric analysis of \(\varphi ^4\) fields and Ising models. Commun. Math. Phys. 86(1), 1–48 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alberici, D., Contucci, P., Mingione, E.: A mean-field monomer–dimer model with attractive interaction: exact solution and rigorous results. J. Math. Phys. 55, 063301 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allegra, N., Fortin, J.: Grassmannian representation of the two-dimensional monomer–dimer model. Phys. Rev. 89, 062107 (2014)CrossRefGoogle Scholar
  4. 4.
    Au-Yang, H., Perk, J.H.H.: Ising correlations at the critical temperature. Phys. Lett. A 104(3), 131–134 (1984)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ayyer, A.: A statistical model of current loops and magnetic monopoles. Math. Phys. Anal. Geom. 18(1), 1–19 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. 181(3), 1087–1138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dubédat, J.: Exact bosonization of the Ising model, arXiv:1112.4399, (2011)
  8. 8.
    Dubédat, J.: Dimers and families of Cauchy–Riemann operators I. J. Am. Math. Soc. 28(4), 1063–1167 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fisher, M.E., Stephenson, J.: Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. 132(4), 1411–1431 (1963)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fisher, M., Hartwig, R.E.: In: Shuler, K.E. (ed.) Stochastic Processes in Chemical Physics, vol. 15, p. 333. Wiley, New York (1969)Google Scholar
  11. 11.
    Giuliani, A., Greenblatt, R.L., Mastropietro, V.: The scaling limit of the energy correlations in non-integrable Ising models. J. Math. Phys. 53, 095214 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giuliani, A., Mastropietro, V., Toninelli, F.: Height fluctuations in non-integrable classical dimers. EPL (Europhys. Lett.) 109(6), 60004 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Giuliani, A., Mastropietro, V., Toninelli, F.L.: Height fluctuations in interacting dimers, Annales de l’Institut Henri Poincaré, Probability and Statistics, in press, arXiv:1406.7710, (2015)
  14. 14.
    Groeneveld, J., Boel, R., Kasteleyn, P.: Correlation-function identities for general planar Ising systems. Physica A 93(1–2), 138–154 (1978)ADSCrossRefGoogle Scholar
  15. 15.
    Hartwig, R.E.: Monomer pair correlations. J. Math. Phys. 7(2), 286–299 (1966)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Heilmann, O.J., Lieb, E.H.: Monomers and dimers. Phys. Rev. Lett. 24(25), 1412–1414 (1970)ADSCrossRefGoogle Scholar
  17. 17.
    Heilmann, O.J., Lieb, E.H.: Theory of monomer–dimer systems. Commun. Math. Phys. 25(3), 190–232 (1972). Errata, Vol. 27, p. 166ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. J. Stat. Phys. 48, 1–2 (1987)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kasteleyn, P.W.: Dimer statistics and phase transitions. J. Math. Phys. 4(2), 287–293 (1963)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kenyon, C., Randall, D., Sinclair, A.: Approximating the number of monomer–dimer coverings of a lattice. J. Stat. Phys. 83, 3–4 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kong, Y.: Monomer–dimer model in two-dimensional rectangular lattices with fixed dimer density. Phys. Rev. E 74, 061102 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    Krauth, W.: Statistical Mechanics: Algorithms and Computations. Oxford Masters Series in Statistical, Computational, and Theoretical Physics. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  25. 25.
    Kuperberg, G.: Symmetries of plane partitions and the permanent-determinant method. J. Comb. Theory Ser. A 68(1), 115–151 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lieb, E.H.: Solution of the dimer problem by the transfer matrix method. J. Math. Phys. 8(12), 2339–2341 (1967)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Lieb, E.H.: A theorem on Pfaffians. J. Comb. Theory 5, 313–319 (1968)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lieb, E.H., Loss, M.: Fluxes, Laplacians, and Kasteleyn’s theorem. Duke Math. J. 71(2), 337–363 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pinson, H., Spencer, T.: Universality and the two-dimensional Ising model, unpublishedGoogle Scholar
  30. 30.
    Priezzhev, V.B., Ruelle, P.: Boundary monomers in the dimer model. Phys. Rev. E 77, 061126 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 333(3), 239–244 (2001)ADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172(2), 1435–1467 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Spencer, T.: A mathematical approach to universality in two dimensions. Physica A 279, 250–259 (2000)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6(68), 1061–1063 (1961)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Temperley, H.N.V.: Enumeration of graphs on a large periodic lattice, combinatorics. In: The Proceedings of the British Combinatorical Conference, 1973, London Mathematical Society lecture note series, Vol. 13, Cambridge University Press, 1974Google Scholar
  36. 36.
    Tzeng, W., Wu, F.Y.: Dimers on a simple-quartic net with a vacancy. J. Stat. Phys. 110(3–6), 671–689 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wu, F.Y.: Pfaffian solution of a dimer–monomer problem: single monomer on the boundary. Phys. Rev. E 74, 020104 (2006). Erratum: n. 039907(E)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Wu, F.Y., Tzeng, W., Izmailian, N.S.: Exact solution of a monomer-dimer problem: a single boundary monomer on a nonbipartite lattice. Phys. Rev. E 83, 011106 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Alessandro Giuliani
    • 1
  • Ian Jauslin
    • 2
  • Elliott H. Lieb
    • 3
  1. 1.Dipartimento di Matematica e FisicaUniversità di Roma TreRomaItaly
  2. 2.Dipartimento di FisicaSapienza Università di RomaRomaItaly
  3. 3.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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