Tricolor percolation and random paths in 3D

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?