Domino tilings of the Aztec diamond with random one-periodic edge weights domino-tilings
Leonid Petrov
Displays colored domino tilings of an Aztec diamond with random one-periodic edge weights. Five parameter regimes (critical, rare event, Bernoulli, periodic, uniform) create varying degrees of disorder. Controls set the diamond size, regime, and toggle domino, dimer, path, and height function views.
This simulation generates random tilings with random edge weights via the shuffling algorithm. The original python code was created by Sunil Chhita; this version is adapted for JS + WebAssembly. The sampler works in your browser. Up to $n \sim 120$ it works in reasonable time, but for larger $n$ it may take a while.
I set the upper bound at $n=400$ to avoid freezing your browser.
Parameter Regimes
Parameters: $1 + \frac{2}{\sqrt{N}}$ with probability $p_1$, and $1 - \frac{1}{\sqrt{N}}$ with probability $p_2$.
Models critical behavior near the uniform measure.
Parameter equals $v_1$ with probability $\frac{1}{\sqrt{N}}$, and $v_2$ with probability $\frac{\sqrt{N} - 1}{\sqrt{N}}$.
Models rare high-weight events in the limit.
Parameter equals $v_1$ with probability $\frac{1}{2}$, and $v_2$ with probability $\frac{1}{2}$.
Two-point discrete distribution.
• With probability Prob 1: weight = Value 1
• With probability Prob 2: weight = Value 2
Simple Bernoulli distribution. Your values and probabilities are used directly.
Deterministic periodic pattern: $w_1, w_2, w_1, w_2, w_1, w_2, \ldots$
Fixed alternating weights with no randomness.
Parameters are independent and uniform on $[a,b]$.
Continuous uniform distribution.
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)