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Local statistics of lattice dimers

Richard Kenyon
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Abstract

We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy $\mu$ on the space of tilings of the plane with dominos. We construct a measure $\nu$ on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the $\nu$-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for $\mu$ and $\nu$, and compute the rate of convergence to mixing (the correlation between distant events). For the measure $\nu$ we compute the variance of the height function.

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Richard Kenyon. 2001-05-08. Local statistics of lattice dimers. https://doi.org/10.1016/s0246-0203(97)80106-9

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