Conformal invariance of domino tiling
Richard Kenyon
Abstract
Let U be a multiply-connected region in R2 with smooth boundary. Let Pepsilon be a polyomino in epsilon Z2 approximating U as epsilon tends to 0. We show that, for certain boundary conditions on Pepsilon, the height distribution on a random domino tiling (dimer covering) of Pepsilon is conformally invariant in the limit as epsilon tends to 0, in the sense that the distribution of heights of boundary components only depends on the conformal type of U. The mean height and all the moments are explicitly evaluated.
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