## Abstract

We study "tricolor percolation" on the regular tessellation of R^{3} by truncated octa-hedra, 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× n×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 SLE_{6} in two dimensions? What can be shown for the higher dimensional analogs of this problem?.

Original language | English |
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Article number | 4 |

Journal | Electronic Journal of Probability |

Volume | 19 |

DOIs | |

State | Published - 6 Jan 2014 |

## Keywords

- Body centered cubic lattice
- Permutahedron
- Tricolor percolation
- Truncated octahedron
- Vortex models