## Conformal invariance of domino tiling.(English)Zbl 1043.52014

Summary: Let $$U$$ be a multiply connected region in $$\mathbb{R}^2$$ with smooth boundary. Let $$P_\varepsilon$$ be a polyomino in $$\varepsilon\mathbb{Z}^2$$ approximating $$U$$ as $$\varepsilon\to 0$$. We show that, for certain boundary conditions on $$P_\varepsilon$$ the height distribution on a random domino tiling (dimer covering) of $$P_\varepsilon$$ conformally invariant in the limit as $$\varepsilon$$ 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.

### MSC:

 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 05B45 Combinatorial aspects of tessellation and tiling problems 05B50 Polyominoes 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

### Keywords:

Domino tiltings; dimer model; conformal invariance
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### References:

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