Holey hexagons with holes at different heights

Leonid Petrov


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.)

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 • (Main GitHub repo)

(note: parameters in the code might differ from the ones in simulation results below)

simulation results

  1. Tiling with a symmetric hole • (data: 57 KB) • (graphics: 8.7 MB)
    Hexagon of size 100, hole size 15, shift 0.
    Tiling with a symmetric hole
  2. Tiling with a skewed hole • (data: 57 KB) • (graphics: 8.7 MB)
    Hexagon of size 100, hole size 15, shift 4.
    Tiling with a skewed hole
  3. Tiling with an even more skewed hole • (data: 57 KB) • (graphics: 8.7 MB)
    Hexagon of size 100, hole size 15, shift 8.
    Tiling with an even more skewed hole