Canvas displaying rainbow-colored non-intersecting lattice paths of the stochastic colored six-vertex model, revealing arctic curve phenomena. Paths enter from the left boundary with distinct colors and interact via probabilistic crossing rules controlled by weights b1 and b2. Adjustable grid size, display settings for line thickness, saturation, and lightness, plus PNG/PDF/SVG export.
Description
The stochastic colored six-vertex model is an 𝔰𝔩n+1-related integrable stochastic system on a square lattice. Each path carries a "color" (an integer label), and at each vertex, two incoming paths (from the left and bottom) interact and produce two outgoing paths (going up and right) according to probabilistic rules.
The stochastic weights depend on the relative ordering of the incoming colors:
- When bottom color > left color: paths cross with probability 1−b2, or go straight with probability b2
- When bottom color < left color: paths go straight with probability b1, or cross with probability 1−b1
- When colors are equal: paths continue straight (deterministic)
The boundary conditions place paths with colors 0, 1, 2, ..., N−1 entering from the left boundary at heights 0, 1, 2, ..., N−1, respectively. The rainbow coloring visualizes how these colored paths interact, cross, and separate as they evolve through the lattice, revealing the characteristic "arctic curve" phenomenon where paths freeze into deterministic regions.
References
- A. Borodin, M. Wheeler, Coloured stochastic vertex models and their spectral theory (opens in new tab), arXiv:1808.01866 [math.PR], 2018.
code
(note: parameters in the code might differ from the ones in simulation results below)-
Link to code(C++ source code for the simulation)