Tricolor percolation and random paths in 3D
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Tricolor percolation and random paths in 3D |
| 2. | Creator | Author's name, affiliation, country | Scott Sheffield; Massachusetts Institute of Technology; United States |
| 2. | Creator | Author's name, affiliation, country | Ariel Yadin; Ben Gurion Univeristy of the Negev; Israel |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | tricolor percolation, vortex models, truncated octahedron, body centered cubic lattice, permutahedron |
| 3. | Subject | Subject classification | 60K35; 82B43 |
| 4. | Description | 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? |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | |
| 7. | Date | (YYYY-MM-DD) | 2014-01-06 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/3073 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v19-3073 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 19 |
| 12. | Language | English=en | en |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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