A variational principle for domino tilings

Details A variational principle for domino tilings A variational principle for domino tilings

2000-08-30

arxiv.org/abs/math/0008220v3

Journal of the AMS 14 (2001), 297-346

Mathematics

Combinatorics

49 pages, 7 figures. This revision of the published paper corrects a typo (reversal of s and t, and an incorrect sign) that pr

Scientific paper

10.1090/S0894-0347-00-00355-6

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within epsilon (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

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