Tricolor percolation and random paths in 3D
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...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89532 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1826213003908874240 |
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| author | Yadin, Ariel Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Yadin, Ariel Sheffield, Scott Roger |
| author_sort | Yadin, Ariel |
| collection | MIT |
| description | 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? |
| first_indexed | 2024-09-23T15:42:07Z |
| format | Article |
| id | mit-1721.1/89532 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:42:07Z |
| publishDate | 2014 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/895322022-09-29T15:33:24Z Tricolor percolation and random paths in 3D Yadin, Ariel Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger 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) 2014-09-15T17:33:43Z 2014-09-15T17:33:43Z 2014-01 2013-12 Article http://purl.org/eprint/type/JournalArticle 1083-6489 http://hdl.handle.net/1721.1/89532 Sheffield, Scott, and Ariel Yadin. “Tricolor Percolation and Random Paths in 3D.” Electronic Journal of Probability 19, no. 0 (January 2, 2014). https://orcid.org/0000-0002-5951-4933 en_US http://dx.doi.org/10.1214/EJP.v19-3073 Electronic Journal of Probability Creative Commons Attribution http://creativecommons.org/licenses/by/3.0/ application/pdf Institute of Mathematical Statistics Institute of Mathematical Statistics |
| spellingShingle | Yadin, Ariel Sheffield, Scott Roger Tricolor percolation and random paths in 3D |
| title | Tricolor percolation and random paths in 3D |
| title_full | Tricolor percolation and random paths in 3D |
| title_fullStr | Tricolor percolation and random paths in 3D |
| title_full_unstemmed | Tricolor percolation and random paths in 3D |
| title_short | Tricolor percolation and random paths in 3D |
| title_sort | tricolor percolation and random paths in 3d |
| url | http://hdl.handle.net/1721.1/89532 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT yadinariel tricolorpercolationandrandompathsin3d AT sheffieldscottroger tricolorpercolationandrandompathsin3d |