Tricolor percolation and random paths in 3D
@article{EJP3073,
author = {Scott Sheffield and Ariel Yadin},
title = {Tricolor percolation and random paths in 3D},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {tricolor percolation, vortex models, truncated octahedron, body centered cubic lattice, permutahedron},
abstract = {We study "tricolor percolation" on the regular tessellation of $\mathbb{R}^3$ by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector $p = (p_1, p_2, p_3)$ and define a "tricolor edge" to be an edge incident to one cell of each color. The tricolor edges form disjoint loops and/or infinite paths. These loops and paths have been studied in the physics literature, but little has been proved mathematically.
We show that each $p$ belongs to either the compact phase (in which the length of the tricolor loop passing through a fixed edge is a.s. finite, with exponentially decaying law) or the extended phase (in which the probability that an $(n \times n \times n)$ box intersects a tricolor path of diameter at least $n$ exceeds a positive constant, independent of $n$). We show that both phases are non-empty and the extended phase is a closed subset of the probability simplex.
We also survey the physics literature and discuss open questions, including the following: Does $p=(1/3,1/3,1/3)$ belong to the extended phase? Is there a.s. an infinite tricolor path for this $p$? Are there infinitely many? Do they scale to Brownian motion? If $p$ lies on the boundary of the extended phase, do the long paths have a scaling limit analogous to SLE6 in two dimensions? What can be shown for the higher dimensional analogs of this problem?
},
pages = {no. 4, 1-23},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3073},
url = {http://ejp.ejpecp.org/article/view/3073}}