# Experimental Evidence of Random Tiling Limit Shapes

In the area of random tilings, there are many results that fall under the term "Arctic Circle Theorems." This roughly means that if one chooses a tiling of a specific region uniformly at random, then there will generally be a boundary region outside which the tiles are "frozen" in a fixed configuration. Here are some computer generated examples:

I keep seeing references in the random tiling literature to models in statistical physics but I've never actually seen a picture from an experiment. I've heard for example that random dimer models correspond to adsorption of a gas onto a thin film. I'd be thrilled to see some pictures of these limit shapes, from say a scanning electron microscope (obviously the tiles involved should be large enough to discern, so perhaps a gas is a bad example). I'm not asking for a bunch of examples, just one or two is more than enough. An example that would fit my criteria is one where the partition function is close to one of a random tiling model, such as dimers or rhombii.

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This is interesting but awfully broad. Do you have a specific question? A list of examples with pictures is out of scope for Physics.SE. – Brandon Enright Mar 2 '14 at 1:15
@BrandonEnright: unfortunately I am not a physicist but I would be happy to narrow this question down with some suggestions. I'm honestly just interested in any particular example where the partition function is more or less the same. I mentioned the random dimer model but I'm not at all sure one can capture it with a picture. – Alex R. Mar 2 '14 at 1:18
Wow, I've never heard of this; to paraphrase G Marx, "That's the interestingest thing I ever saw!". What do you mean, "random tiling": are the tiles all the same shape and we simply choose their colour at random as they are laid down? I don't think I understand this at all. Intuitively, it would seem that nearest neighbour colours are uncorrelated, so how can seemingly correlated regions form? I'm not disputing you: I'm just checking I understand right. Can you give a simple reference as to how this happens? – WetSavannaAnimal aka Rod Vance Mar 3 '14 at 0:32
@WetSavannaAnimalakaRodVance: in suresh's answer below, the reference to one or Okounkovs lectures provides a nice exposition. In the above example of the diamond, there are actually only two types of tiles, vertical and horizontal but the colors correspond to something called "height functions" which form the backbone of analyzing the tilings. In a way, you can try to zoom into one or the diamond frozen regions above, and try to displace a single frozen tile, say from horizontal to vertical. You'll quickly see that to retain a tiling you have to change a significant number or adjacent tiles. – Alex R. Mar 3 '14 at 2:00
@WetSavannaAnimalakaRodVance: also to clarify, a key ingredient is that the tiling is uniformly random. In other words out of all possible allowable tilings, we select one uniformly at random. – Alex R. Mar 3 '14 at 2:06