"Coarse" Arctic Circle Theorem - MathOverflow

"Coarse" Arctic Circle Theorem

2011-10-18T14:00:03Z

Uniformly random dimer tilings of aztec diamond region will have an "artic" circle appear in the middle.

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What happens if we make the regions a little bit coarser, do we get the same limit shape?

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I apologize for the lousy aspect ratio. In general, does the limit shape depend only on the coutour or does it depend "microscopically" on the boundary conditions as well?

Answer by Joseph O'Rourke for "Coarse" Arctic Circle Theorem

2011-10-18T14:35:52Z

The limit shape definitely depends on the contour. Richard Kenyon's "Lectures on dimers" (PDF link) explains the status (p.32):

For general boundary conditions ﬁnding the analytic function $Q$ in [equation] (11) which describes the limit shape is diﬃcult. For boundary conditions resembling those in Figure 4, however, one can give an explicit answer.

The above only makes full sense in the context of his notes, but he can compute the limit shape in the case below, where it is a cardiod:

I believe that the limit shape can be computed for all polygonal approximations of a certain type, those "resembling those in Figure 4."

Answer by Henry Cohn for "Coarse" Arctic Circle Theorem

2011-10-18T15:55:25Z

To expand slightly on Joseph's answer and Noam's comment, you can describe a domino or lozenge tiling by its "height function." In the case of lozenge tilings, shown in Joseph's answer, the height function is visible, since the picture can naturally be interpreted as showing a 3d surface. (You have to make one choice, since there's an ambiguity regarding what is pointing into the screen and what is pointing out.) For domino tilings, it is not nearly as visually apparent, but height functions are explained in a beautiful paper by Thurston (Conway's tiling groups, Amer. Math. Monthly 97 (1990), 757–773).

The microscopic shape of a region influences the height function, since it determines the wire-frame boundary the height function must interpolate. For example, for Aztec diamonds the boundary is as steep as possible, while in the coarser example it stays flat.

It turns out that in the continuum limit, the shape of a typical tiling is entirely determined by the shape of the boundary, and it is determined by a variational principle: among all spanning surfaces that satisfy a certain Lipschitz constraint (that must hold for all tilings), the typical surface is singled out by maximizing the entropy, which is given by integrating local contributions that can be computed explicitly in terms of the partial derivatives of the height function. See http://front.math.ucdavis.edu/math.CO/0008220 for more details. Almost all tilings will cluster around the typical tiling, so this completely determines the large-scale behavior.

The two examples from the question can be completely analyzed in this framework. In particular, the second one has the highest possible entropy at every point and therefore has nothing like an Arctic circle. See Figure 4 in http://front.math.ucdavis.edu/math.CO/0008243 (reproduced below) for a picture of a random tiling.

Kenyon, Okounkov, and Sheffield (http://front.math.ucdavis.edu/0311.4705) substantially generalized this framework, and Kenyon and Okounkov (http://front.math.ucdavis.edu/0507.4707) found a beautiful description of the variational problem for lozenge tilings in terms of the complex Burgers equation. This leads to explicit descriptions of cases like the cardioid in Joseph's answer.