Glauber Dynamics on Domino tilings with 2x2 periodic weights     domino-tilings

Colored domino tiling of a diamond-shaped region rendered as an SVG mosaic. Demonstrates Glauber dynamics on Aztec diamond tilings with 2-by-2 periodic edge weights (a,b). The Markov chain flips adjacent domino pairs; adjusting weights in real time reveals arctic circle phenomena and limit shape transitions.

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.

Skip to simulation canvas

code

• 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)

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