At each vertex of a domain of the square lattice of the square lattice one assigns an arrow to the adjacent edges (either up/down or left/right) such that the number of arrows pointing into the vertex equals the number pointing out. This constraint limits the number of configurations at a vertex to six, giving the model its name. Below are the six possible vertices drawn as both arrows and paths.
We will consider an \(N\times N\) square with domain wall boundary conditions, that is, the top and right boundaries are fully packed with paths while the bottom and left are empty.
A randomly sampled configuration of the DWBC six vertex model of size \(N=250\) with \(a=1,b=2,c=2\).
The derivative of the height function with \(N=250\) and \(a=1,b=2,c=2\) averaged over 200 configurations. The curve in red is the theoretically predicted arctic curve.
The derivative of the height function of a single configuration with \(N=200\) and \(a=b=1,c=\sqrt{8}\).
Some tilings of generalized tower graphs. Made with the shuffling algorithm described in this article.
A tiling of the Aztec diamond of rank 20. Another Aztec diamond of rank 200.
A tiling of the tower graph of rank 20. Another tower graph of rank 200.
A tiling of the rank-20 generalized tower graph whose boundary approximates the function \(f(x)=\frac{1}{3}x^2 +\frac{2}{3}\). Another tiling of rank 200.
A tiling of the rank-20 generalized tower graph whose boundary approximates the function \(f(x)=\frac{1}{3\pi} \sin(2\pi x)+1\). Another tiling of rank 200.
A tiling of the rank-20 generalized tower graph whose boundary approximates the function \(f(x)=\frac{\tan^{-1}(3x)+\tan^{-1}(3)}{3} +1\). Another tiling of rank 200.
A small tiling whose boundary is given by a random path. A large tiling whose boundary is given by a random path.
We consider pairs of tilings of the Aztec diamond, color one blue and the other red. We think of the tilings as being superimposed one on top of the other with an interaction whenever we see a pair of coupled dominos, that is, overlapping dominos of the form
We assign a weight of \(t^{\text{# coupled pairs}}\) to each 2-tiling. See this article for where the coupled tilings are defined. The tilings are generated using the shuffling algorithm from this article.
The arctic curve of one of the rank-128 tilings of a 2-tiling when the \(t=0\). (The arctic curve is the same for the other tiling.)
A rank-128 2-tiling with \(t=0.2\) drawn as a collection of loops and double edges.
A rank-256 2-tiling with \(t=0.2\) drawn as a collection of loops and double edges.
Stochastic Box Ball System
We consider a faulty box-ball system in which there is probability \(\varepsilon_p\) that a ball is not picked up by the carrier and probability \(\varepsilon_d\) that the ball is not dropped from the carrier. Define a vector of inter-ball spacings as shown below:
In the above, we have a 4-ball system with spacing vector \(X=(X_1,X_2,X_3)\). The case \(\varepsilon_d=0\) was studied in this article.
The spacings in a 3-ball system with \(\varepsilon_p=\frac{1}{2},\varepsilon_d=0\) ran for \(10,000\) steps.
The spacings in a 4-ball system with \(\varepsilon_p=\frac{1}{2},\varepsilon_d=0\) ran for \(10,000\) steps.
The spacings in a 3-ball system with \(\varepsilon_p=\frac{1}{2},\varepsilon_d=\frac{1}{2}\) ran for \(10,000\) steps.
The spacings in a 3-ball system with \(\varepsilon_p=0,\varepsilon_d=\frac{1}{2}\) ran for \(10,000\) steps.
Oscillating Tableaux
A simulation of a large oscillating tableaux. Generated via a hook-walk algorithm described in this article.
Bounded Lecture Hall Tableaux
Some simulations of bounded lecture hall tableaux. See this article.
