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A variational principle for domino tilings (2001)
| Content Provider | CiteSeerX |
|---|---|
| Author | Cohn, Henry Kenyon, Richard Propp, James Kasteleyn, Willem |
| Abstract | Abstract. 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 ε (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. The effect of boundary conditions is, however, not entirely trivial and will be discussed in more detail in a subsequent paper. P. W. Kasteleyn, 1961 1. |
| File Format | |
| Journal | J. Amer. Math. Soc |
| Language | English |
| Publisher Date | 2001-01-01 |
| Access Restriction | Open |
| Subject Keyword | Variational Principle Domino Tiling Dimer Problem Unique Entropy-maximizing Solution Square Grid Typical Tiling Arbitrary Finite Region Boundary Condition Random Domino Tiling Tiling Model Unit Lozenge Local Behavior Domino Tiling Dimer Model Subsequent Paper General Boundary Condition |
| Content Type | Text |
| Resource Type | Article |
