Local statistics of lattice dimers

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 $μ$ on the space of tilings of the plane with dominos. We construct a measure $ν$ on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the $ν$-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for $μ$ and $ν$, and compute the rate of convergence to mixing (the correlation between distant events). For the measure $ν$ we compute the variance of the height function.

27 pages, 6 figures