Continuing with the setup of the previous simulation (see here), we consider a simple, deterministic transformation of the tiling, when at each step on one of the levels the vertical lozenges are mirrored with respect to the middle of the segment [min,max] to which each of them belong. The dynamics starts from an exact sample of the measure $q^{-volume}$ (which is produced by Vadim Gorin’s program [2]). Then, by mirroring vetical lozenges to the left, the measure $q^{-volume}$ becomes the measure $q^{volume}$. At each step, the distribution of the tiling is the same as in the previous simulation with random moves. However, under the mirroring dynamics the vertical tiles can move both left and right, and the Markovian nature of the behavior on the left edge is lost.
Here are three states of the tiling in the beginning, the middle of the simulation, and the end. We see that the beginning and the ending configuration are exactly the same, up to reflection about the center of the hexagon.
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 \(\{ \{ 47 \},\{ 50,47 \} , \{ 50,49,47 \} ,\ldots, \} .\) Here $t$ is the time variable. The simulation data can be “coarse” in larger tilings, or finer with every step of the Markov chain recorded.
https://github.com/lenis2000/simulations/tree/master/2019-05-01-qvol-sampler
(python code for simulations, simple Mathematica code for drawing)
https://arxiv.org/abs/0804.3071
• https://www.sciencedirect.com/science/article/pii/S0001870808003253
https://arxiv.org/abs/1712.04584