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.)
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.
https://github.com/lenis2000/Glauber_Simulation/tree/holey-hexagons
(python code (Glauber2.py) for simulations, simple Mathematica code (Dynamic.nb) for drawing)