A variational principle for domino tilings

Cohn, Henry^1, 2 ; Kenyon, Richard³ ; Propp, James⁴

¹ Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
² Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
³ CNRS UMR 8628, Laboratoire de Topologie, Bâtiment 425, Université Paris-11, 91405 Orsay, France
⁴ Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Journal of the American Mathematical Society, Tome 14 (2001) no. 2, pp. 297-346

Résumé

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 $\varepsilon$ (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.

Détail

DOI : 10.1090/S0894-0347-00-00355-6

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