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Local statistics of lattice dimers. (English) Zbl 0893.60047

Summary: 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 distinct events). For the measure \(\nu\) we compute the variance of the height function.

MSC:

60J35 Transition functions, generators and resolvents
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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