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?
