Random rhombus tilings

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Joakim Linde, Cristopher Moore and Mats G. Nordahl (2001), An n-Dimensional Generalization of the Rhombus Tiling, Discrete Mathematics and Theoretical Computer Science Proceedings AA, pp. 23-42.




Examples of random rhombus tilings

Click on the thumbnails to see larger versions of the images.

Shape of the average height function

After the DMTCS paper some measurements have been done regarding the shape of the average height function for rhombus tilings.

average height function of 2d rhombus tilings
2d: The height function has been rescaled so that it is independent of system size. The measured results almost coincides for all examined system sizes. The measured results are also in close agreement with the analytical results for tilings of infinite size (H. Cohn, M. Larsen and J. Propp (1998), The Shape of a Typical Boxed Plane Partition, New York J. of Math. 4, 137).

average height function of 3d rhombus tilings
3d: The height function has been rescaled so that it is independent of system size. It has been conjectured that the height function is flat in the unfrozen region. From the measurements shown above it looks like the height function is almost, but not quite, flat. If that is due to a thus far undiscovered measurement bug, or that the height function actually isn't quite flat, is unknown.