Dynamics on q-vol lozenge tilings inverting the parameter q

Leonid Petrov and Edith Zhang


The model [1]

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 $q^{volume}$.

Sampling algorithm

The simulation is based on a bijectivisation of the Yang-Baxter equation [3]. Its details are explained in the forthcoming publication [1].

Pictorial results

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).

Configuration in the beginning

Configuration in the middle

Configuration in the end

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 $\{ \{ 47 \},\{ 50,47 \} , \{ 50,49,47 \} ,\ldots, \} .$ 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.


code • (Main GitHub repo)

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

simulation results

  1. A gif example of a dynamics of only vertical lozenges • (graphics: 18 MB)
    Hexagon $10\times 10\times 10$, $q=0.85$
    A gif example of a dynamics of only vertical lozenges
  2. GIF of a dynamics of only vertical lozenges • (graphics: 18 MB)
    Hexagon $10\times 10\times 10$, $q=0.85$
    GIF of a dynamics of only vertical lozenges
  3. Movie, size 10, all frames • (graphics: 300 KB)
    Hexagon $10\times 10\times 10$, $q=0.85$.
    Movie, size 10, all frames
  4. Movie, size 25, all frames. Almost frozen configuration • (graphics: 1.2 MB)
    Hexagon $25\times 25\times 25$, $q=0.7$.
    Movie, size 25, all frames. Almost frozen configuration
  5. Movie, size 50, only frames after each sweep • (data: 1.3 MB) • (graphics: 2.4 MB)
    Hexagon $50\times 50\times 50$, $q=0.95$.
    Movie, size 50, only frames after each sweep
  6. Movie, size 50, all frames • (data: 63.6 MB) • (graphics: 7.3 MB)
    Hexagon $50\times 50\times 50$, $q=0.95$. Took several days to render all the 5K frames
    Movie, size 50, all frames
  7. Movie, size 100, only frames after each sweep • (data: 11.2 MB) • (graphics: 24.1 MB)
    Hexagon $100\times 100\times 100$, $q=0.95$.
    Movie, size 100, only frames after each sweep

references

  1. L. Petrov, A. Saenz. In preparation (2019)
  2. A.Borodin, V. Gorin. Shuffling algorithm for boxed plane partitions. Advances in Mathematics, 220 (6) (2009). 1739-1770, • https://arxiv.org/abs/0804.3071https://www.sciencedirect.com/science/article/pii/S0001870808003253
  3. A. Bufetov, L. Petrov. Yang-Baxter field for spin Hall-Littlewood symmetric functions (2017) • https://arxiv.org/abs/1712.04584