Electronic Journal of Probability

Tricolor percolation and random paths in 3D

Scott Sheffield and Ariel Yadin

Full-text: Open access

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?

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 4, 23 pp.

Dates
Accepted: 6 January 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065646

Digital Object Identifier
doi:10.1214/EJP.v19-3073

Mathematical Reviews number (MathSciNet)
MR3164757

Zentralblatt MATH identifier
1307.60147

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
tricolor percolation vortex models truncated octahedron body centered cubic lattice permutahedron

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Sheffield, Scott; Yadin, Ariel. Tricolor percolation and random paths in 3D. Electron. J. Probab. 19 (2014), paper no. 4, 23 pp. doi:10.1214/EJP.v19-3073. https://projecteuclid.org/euclid.ejp/1465065646


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