Tricolor percolation and random paths in 3D

Scott Sheffield, Ariel Yadin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


We study "tricolor percolation" on the regular tessellation of R3 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 = (p1,p2,p3) 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 SLE6 in two dimensions? What can be shown for the higher dimensional analogs of this problem?.

Original languageEnglish
Article number4
JournalElectronic Journal of Probability
StatePublished - 6 Jan 2014


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


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