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.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
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.
Mathematical Reviews (MathSciNet):
MR125252
[3] Cardy, J. L. (1986). Operator content of two-dimensional conformally invariant theories. Nuclear Phys. B 270 186–204.
Mathematical Reviews (MathSciNet):
MR845940
[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.
Mathematical Reviews (MathSciNet):
MR923852
[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.
Mathematical Reviews (MathSciNet):
MR253689
[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.
Mathematical Reviews (MathSciNet):
MR926363
[16] Temperley, H. N. V. and Fisher, M. E. (1961). Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6 1061–1063.
Mathematical Reviews (MathSciNet):
MR136398
[17] Thurston, W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97 757–773.
