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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
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