We consider the measure on lozenge tilings of a hexagon in which the probabilistic weight of every tiling is proportional to $q^{volume}$ or $q^{-volume}$, where $0< q <1$. The dynamics starts from an exact sample of the measure $q^{-volume}$ (which is produced by Vadim Gorin’s program [2]). Then, by randomly moving vetical lozenges to the left, the measure $q^{-volume}$ becomes the measure $q^{volume}$.
The simulation is based on a bijectivisation of the Yang-Baxter equation [3]. Its details are explained in the forthcoming publication [1].
Here is a sample of three random tilings of the hexagon of size $50\times 50\times 50$ in the beginning, midway, and in the end of the simulation. Throughout the simulation, each random configuration has a distribution which is of $q$-Gibbs type, but with powers of $q$ reshuffled (these are the parameters in the corresponding skew Schur factors).
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.
The simulations are gifs or movies.
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