About the simulation

Shuffling (initial picture)

This simulation demonstrates random domino tilings of an Aztec diamond—a diamond‑shaped union of unit squares. The probability measure is $2\times2$‑periodic with edge‑weights $(a,b)$, as studied by Chhita & Johansson in Domino tilings of the Aztec diamond with periodic weights. Sampling uses the shuffling algorithm. The original Python implementation by Sunil Chhita has been ported to JavaScript + WebAssembly, and the graphics are rendered with D3.js.

The sampling runs entirely in your browser. For sizes up to about $n\le120$ the sampler is fast; larger $n$ may take noticeable time (hard cap $n=300$ to protect your browser).

Glauber Dynamics

You can run the Glauber dynamics on domino tilings, and adjust the speed. You can start the dynamics with one set of parameters $(a,b)$ and change them on the fly, observing in real time how the tiling reacts. Key phenomena visible in the grayscale view:

• When $a=b$, the measure is uniform and inside the arctic circle you can see a “liquid” mixture of colors.
• When $a<1, b=1$, lighter color dominates; when $a>1, b=1$, darker color dominates.
• Local color relaxation occurs much faster than changes in the macroscopic limit shape.

Conjecture: In the non‑uniform case $a\neq b$, the Glauber chain requires exponentially many sweeps in $n$ to alter the limit shape.

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code

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

• Link to code (This simulation is interactive, written in JavaScript; see the source code of this page at the link)
• Link to code (C++ code for the simulation)