The Annals of Probability

Loop statistics in the toroidal honeycomb dimer model

Cédric Boutillier and Béatrice de Tilière

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Abstract

The dimer model on a graph embedded in the torus can be interpreted as a collection of random self-avoiding loops. In this paper, we consider the uniform toroidal honeycomb dimer model. We prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed, the winding number of the collection of loops converges in law to a two-dimensional discrete Gaussian distribution. This is known to physicists in more generality from their analysis of toroidal two-dimensional critical loop models and their mapping to the massless free field on the torus. This paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model.

Article information

Source
Ann. Probab., Volume 37, Number 5 (2009), 1747-1777.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1253539856

Digital Object Identifier
doi:10.1214/09-AOP453

Mathematical Reviews number (MathSciNet)
MR2561433

Zentralblatt MATH identifier
1179.60065

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Dimers winding number loop ensemble

Citation

Boutillier, Cédric; de Tilière, Béatrice. Loop statistics in the toroidal honeycomb dimer model. Ann. Probab. 37 (2009), no. 5, 1747--1777. doi:10.1214/09-AOP453. https://projecteuclid.org/euclid.aop/1253539856


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References

  • [1] Alet, F., Jacobsen, J. L., Misguich, G., Pasquier, V., Mila, F. and Troyer, M. (2005). Interacting classical dimers on the square lattice. Phys Rev Lett. 94 235702–235706.
  • [2] Bellman, R. (1961). A Brief Introduction to Theta Functions. Holt, New York.
  • [3] Cardy, J. L. (1986). Operator content of two-dimensional conformally invariant theories. Nuclear Phys. B 270 186–204.
  • [4] Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346 (electronic).
  • [5] de Tilière, B. (2007). Scaling limit of isoradial dimer models & the case of triangular quadri-tilings. Ann. Inst. H. Poincaré Probab. Statist. 43 729–750.
  • [6] Di Francesco, P., Saleur, H. and Zuber, J.-B. (1987). Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models. J. Statist. Phys. 49 57–79.
  • [7] Ferdinand, A. E. (1967). Statistical mechanics of dimers on a quadratic lattice. J. Math. Phys. 8 2332.
  • [8] Kasteleyn, P. W. (1961). The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice. Physica 27 1209–1225.
  • [9] Kasteleyn, P. W. (1967). Graph theory and crystal physics. In Graph Theory and Theoretical Physics 43–110. Academic Press, London.
  • [10] Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759–795.
  • [11] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128–1137.
  • [12] Kenyon, R. Height fluctuations in the honeycomb dimer model. Comm. Math. Phys. 281 675–709.
  • [13] Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056.
  • [14] Kenyon, R. and Wilson, D. B. Conformal radii of loop models. Unpublished.
  • [15] Pasquier, V. (1987). Lattice derivation of modular invariant partition functions on the torus. J. Phys. A 20 L1229–L1237.
  • [16] Temperley, H. N. V. and Fisher, M. E. (1961). Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6 1061–1063.
  • [17] Thurston, W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97 757–773.