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Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in this answer, one can consider higher dimensional zonotopes tiled by rhombic polytopes. I've seen these usually denoted as $D\to d$ tilings, or $d$-dimensional, codimension $D-d$ tilings. We can call a region in such a tiling effectively $D'$ dimensional, if it looks locally like a $D'\to d$ tiling.

In "An n-dimensional generalization of the rhombus tiling" by J. Linde, C. Moore, and M.G. Nordahl, it is conjectured that the octahedron inscribed inside a rhombic dodecahedron plays the same role for $4\to 3$ tilings as the arctic circle does for $3\to 2$ tilings. Outside this octahedron, the tiling is most likely frozen, i.e. locally periodic and with vanishing entropy, while inside the octahedron we have the only region which is effectively $4$ dimensional. However it is also conjectured that the entropy inside the arctic octahedron approaches a uniform distribution, unlike the 2-dimensional case where the entropy peaks at the center of the arctic ellipse.

What is the status of these conjectures? Is it expected that for all zonotope tilings ($d>2$) the arctic regions are polyhedral? Is the entropy expected to be constant inside the arctic region? Is there a conceptual reason why the behaviour changes so drastically from planar tilings to higher dimensions?

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up vote 10 down vote accepted

Basically nothing is known beyond what you've said, and it doesn't look like that's going to change anytime soon.

There is at least one piece of relevant numerical work which I know of: J. Linde, the first author in the paper you mentioned, posted some graphs on his website which suggest that the height function isn't quite flat inside the octahedron. This means that the entropy isn't likely to be constant in there, either.


There are some heuristic reasons why we should expect flat facets on the arctic regions for these problems when d>3. For instance, similar problems have been studied with torus boundary conditions and a slightly different lattice, and the height function there is extremely flat. See http://arxiv.org/abs/1005.4636.

Other than that, the only other work I know about these higher dimensional tilings is numerical work on solid partitions, counted by volume. MacMahon made some guesses about how many solid partitions there are of a given volume, which Knuth proved to be incorrect in the sixties. It seems that MacMahon might have guessed an asymptotically correct answer, however, according to the results of this project. It uses massively parallel computation to count solid partitions by volume:


and their project has generated a paper,


which I just found and haven't read properly yet. There's also some approximate counting work due to Mustonen and Rajesh, which reaches the same conclusion; it's cited in the above paper.

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