What happens if we sample a uniformly random tiling of a hexagon with a hole, but place the hole at different heights? (Thanks to MR for the request to sample these examples.)
Update 2025-05-05: Well, since I am running the Glauber dynamics to sample these pictures, the pictures are not having fluctuations in the height of the hole. So, one can think of these pictures as samples from the fixed filling numbers ensembles. The “shift” in the data is what exactly corresponds to the height of the hole.
Data file format
The data file is a list of lists of lists in Mathematica-readable format, of the form \(\{ \lambda(1),\lambda(2),\ldots,\lambda(T) \},\) where each $\lambda(t)$ is a list of weakly interlacing integer coordinates of the form \(\{ \{ 47,0,0,0,\ldots \},\{ 50,47,0,0,\ldots \} , \{ 50,49,47,0,\ldots \} ,\ldots, \} .\) This list is a square array, and each $\lambda(i)$ is appended by zeroes.
code
(note: parameters in the code might differ from the ones in simulation results below)-
Link to code(python code (Glauber2.py) for simulations, simple Mathematica code (Dynamic.nb) for drawing)
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Tiling with a symmetric hole • (data: 57 KB) • (graphics: 8.7 MB)
Hexagon of size 100, hole size 15, shift 0.
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Tiling with a skewed hole • (data: 57 KB) • (graphics: 8.7 MB)
Hexagon of size 100, hole size 15, shift 4.
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Tiling with an even more skewed hole • (data: 57 KB) • (graphics: 8.7 MB)
Hexagon of size 100, hole size 15, shift 8.
