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A variational principle for domino tilings

Authors: Henry Cohn, Richard Kenyon and James Propp
Journal: J. Amer. Math. Soc. 14 (2001), 297-346
MSC (2000): Primary 82B20, 82B23, 82B30
Published electronically: November 3, 2000
MathSciNet review: 1815214
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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.

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Additional Information

Henry Cohn
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399

Richard Kenyon
Affiliation: CNRS UMR 8628, Laboratoire de Topologie, Bâtiment 425, Université Paris-11, 91405 Orsay, France

James Propp
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Keywords: Random tiling, dominos, variational principle, matchings, dimer model
Received by editor(s): January 12, 1999
Received by editor(s) in revised form: August 11, 2000
Published electronically: November 3, 2000
Additional Notes: The first author was supported by an NSF Graduate Research Fellowship. The third author was supported by NSA grant MDA904-92-H-3060 and NSF grant DMS92-06374, and by a grant from the MIT Class of 1922.
Dedicated: Dedicated to Pieter Willem Kasteleyn (1924–1996)
Article copyright: © Copyright 2000 American Mathematical Society

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