## The Annals of Probability

### Conformal invariance of domino tiling

Richard Kenyon

#### 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 in Project Euclid: 18 April 2002

http://projecteuclid.org/euclid.aop/1019160260

Digital Object Identifier
doi:10.1214/aop/1019160260

Mathematical Reviews number (MathSciNet)
MR1782431

Zentralblatt MATH identifier
01905938

#### Citation

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

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