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.Comment: 27 pages, 6 figure