Simulation Gallery
Six Vertex Model
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 \(a=1,b=2,c=2\).
- Here we draw only the \(c\)-vertices.
- 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 \(a=b=1,c=\sqrt{8}\).
- The gaseous facet of the above image.
Tilings
Coupled Tiling
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 overlapping dominos of the form
We assign a weight of \(t^{\text{# of interactions}}\) to each 2-tiling.
- A 2-tiling of the Aztec diamond with \(t=0.01\).
- Another 2-tiling with \(t=10\).
- A larger, rank 256, 2-tiling with \(t=5\).
- 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.
Box Ball
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 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
Bounded Lecture Hall Tableaux
Ising Model
