## “Coarse” Arctic Circle Theorem

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?

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There's certainly some "microscopic" dependence, because just removing one of the two central rows of the first picture produces a similar-looking shape with only one domino tiling! But this effect is accounted for by the "height function" of the boundary. – Noam D. Elkies Oct 18 2011 at 15:01
My point is both shapes necome squares rotat ed 45° but maybe height function in scaling limit goes from 0 to 1 in first case and stays at 0 in the second. So boundary conditions are different? So inside limit height depends on microscopic conditions on boundary – John Mangual Oct 18 2011 at 23:37

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.

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Henry: You should probably clarify a point, unless I am mistaken: The Kenyon-Okounkov solution applies to the rhombus tiling situation, and I don't think KO have published how to modify it for domino tilings. Other than that, nice answer! – David Speyer Oct 18 2011 at 16:04
Good point, thanks! – Henry Cohn Oct 18 2011 at 16:08
The "highest possible entropy at every point" means that the logarithm of the number of tilings of the coarse aztec diamond (just as for a square of even side or a region with "soft" boundary conditions) is asymptotically the area times $C/\pi$, where $C = 1-1/9+1/25-\dots$ is Catalan's constant, right? – Johan Wästlund Oct 18 2011 at 16:23
And to see what the height function is, we just chess-board color the region and walk around the boundary, keeping track of the difference in the number of black and white pieces of boundary we have seen. For the coarse aztec diamond this difference is bounded, so the height function is essentially constant. In contrast, for the ordinary aztec diamond, two sides are entirely black and two sides entirely white. – Johan Wästlund Oct 18 2011 at 16:26
@Henry: I took the liberty of adding Fig.4 to your post. – Joseph O'Rourke Oct 18 2011 at 19:32

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."

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Nice! As an aside, I admire your posts (questions and answers) on MO a lot for their clarity and the occasional cool graphics. – Nilima Nigam Oct 18 2011 at 15:18
:-) $\mbox{} \mbox{}$ – Joseph O'Rourke Oct 18 2011 at 16:07
Quite beautiful! – J. M. Oct 20 2011 at 5:04