Tricolor percolation and random paths in 3D

• Main Authors: Yadin, Ariel (Author), Sheffield, Scott Roger (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Institute of Mathematical Statistics, 2014-09-15T17:33:43Z.
• Subjects: Article
• Online Access: Get fulltext

We study "tricolor percolation" on the regular tessellation of R[superscript 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 = (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?
United States-Israel Binational Science Foundation (Grant 2010357)
National Science Foundation (U.S.) (Grant DMS 064558)
National Science Foundation (U.S.) (Grant 1209044)