The Annals of Probability

Conformal invariance of domino tiling

Richard Kenyon

Full-text: Open access

Abstract

Let $U$ be a multiply connected region in $\mathbf{R}^ 2$ with smooth boundary. Let $P_\epsilon$ be a polyomino in $\epsilon\mathbf{Z}^2$ approximating $U$ as $\epsilon \to 0$.We show that, for certain boundary conditions on $P_\eqsilon$, the height distribution on a random domino tiling (dimer covering) of $P_\eqsilon$ is conformally invariant in the limit as $\epsilon$ tends to 0, in the sense that the distribution of heights of boundary components (or rather, the difference of the heights from their mean values) only depends on the conformal type of $U$. The mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way. The mean height and all the moments are explicitly evaluated.

Article information

Source
Ann. Probab. Volume 28, Number 2 (2000), 759-795.

Dates
First available: 18 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1019160260

Mathematical Reviews number (MathSciNet)
MR1782431

Digital Object Identifier
doi:10.1214/aop/1019160260

Zentralblatt MATH identifier
01905938

Subjects
Primary: 81T40: Two-dimensional field theories, conformal field theories, etc. 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 30C20: Conformal mappings of special domains

Keywords
Domino tilings dimer model conformal invariance

Citation

Kenyon, Richard. Conformal invariance of domino tiling. The Annals of Probability 28 (2000), no. 2, 759--795. doi:10.1214/aop/1019160260. http://projecteuclid.org/euclid.aop/1019160260.


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References

  • [1] Belavin, A., Polyakov, A. and Zamolodchikov, A. (1984). Infinite conformal symmetry in two-dimensional quantum field theory Nuclear Phys. B 241 333.
  • [2] Benjamini, I. and Schramm, O. (1998). Conformal invariance of Voronoi percolation. Comm. Math. Phys. 197 75-107.
  • [3] Billingsley, P. (1979). Probability and Measure. Wiley, New York.
  • [4] Bl ¨ote, H. W. J. and Hilhorst, H. J. (1982). Roughening transitions and the zerotemperature triangular Ising antiferromagnet. J. Phys. A 15 L631.
  • [5] Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329- 1371.
  • [6] Cardy, J. (1987). Conformal invariance. In Phase Transitions and Critical Phenomena (C. Domb and J. L. Lebowitz, eds.) 11 55-126. Academic Press, New York.
  • [7] Cohn, H, Kenyon, R. and Propp, J. (1999). A variational principle for domino tilings. J. Amer. Math. Soc. To appear.
  • [8] Doyle, P. and Snell, J. L. (1984). RandomWalks and Electrical Networks. Math. Assoc. of America, Washington, D.C.
  • [9] Duffin, R. J. (1956). Basic properties of discrete analytic functions. Duke Math. J. 23 335- 363.
  • [10] Fournier, J.-C. (1995). Pavage des figures planes sans trous par des dominos: fondement graphique de l'algorithm de Thurston et parallelisation. C. R. Acad. Sci. S´er. I 320 107-112.
  • [11] Guttmann, A. and Bursill, R. (1990). Critical exponent for the loop-erased self-avoiding walk by Monte Carlo methods. J. Statist. Phys. 59 1-9.
  • [12] Kasteleyn, P. (1961). The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27 1209-1225.
  • [13] Kenyon, R. (1997). Local statistics of lattice dimers. Ann. Inst. H. Poincar´e Probab. Statist. 33 591-618.
  • [14] Kenyon, R., Propp, J. and Wilson, D. (2000). Trees and matchings. Electron. J. Combin. 7 Research Paper 25.
  • [15] Kondev, J. and Henley, C. (1995). Geometrical exponents of contour loops on random Gaussian surfaces. Phys. Rev. Lett. 74 4580-4583.
  • [16] St ¨ohr, A. (1954). ¨Uber einige lineare partielle Differenzengleichungen mit konstanter Koeffizienten III. Math. Nachr. 3 330-357.
  • [17] Temperley, H. (1974). Combinatorics: Proceedings of the British Combinatorial Conference 1973. 202-204. Cambridge Univ. Press.
  • [18] Tesler, G. (2000). Matchings in graphs on non-oriented surfaces. J. Combin. Theory Ser. B 78 198-231.
  • [19] Thurston, W. P. (1990). Conway's tiling groups. Amer. Math. Monthly 97 757-773.