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April 2000 Conformal invariance of domino tiling
Richard Kenyon
Ann. Probab. 28(2): 759-795 (April 2000). DOI: 10.1214/aop/1019160260

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.

Citation

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Richard Kenyon. "Conformal invariance of domino tiling." Ann. Probab. 28 (2) 759 - 795, April 2000. https://doi.org/10.1214/aop/1019160260

Information

Published: April 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1043.52014
MathSciNet: MR1782431
Digital Object Identifier: 10.1214/aop/1019160260

Subjects:
Primary: 05A15, 05B45, 30C20, 81T40

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 2 • April 2000
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