Tricolor percolation and random paths in 3D
Ariel Yadin (Ben Gurion Univeristy of the Negev)
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?
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-23
Publication Date: January 6, 2014
DOI: 10.1214/EJP.v19-3073
References
- M. Aizenman, and D.J. Barsky. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489-526. MR0874906
- R.M. Bradley, P. N. Strenski, and J.-M. Debierre. Surfaces of percolation clusters in three dimensions. Physical Review B, 44 (1991), 76-84.
- R.M. Bradley, J.-M. Debierre, and P.N. Strenski. Anomalous scaling behavior in percolation with three colors. Physical review letters, 68(15) (1992), 2332-2335.
- R. M. Bradley, P.N. Strenski, and J.-M. Debierre. A growing self-avoiding walk in three dimensions and its relation to percolation. Physical Review A, 45(12) (1992), p.8513.
- R.M. Bradley, J.-M. Debierre, and P.N. Stenski. A novel growing self-avoiding walk in three dimensions. Journal of Physics A, 25(9) (1992) p.L541.
- R.M. Burton, and M. Keane. Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989), no. 3, 501-505. MR0990777
- F. Camia, and C. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys. 268 (2006), no. 1, 1-38. MR2249794
- D. S. Gaunt, and M. F. Sykes. Series study of random percolation in three dimensions. Journal of Physics A, 16 (1983), 783-799.
- Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339
- M. Hindmarsh, and K. Strobl. Statistical properties of strings. Nuclear Physics B, 437(2) (1995), 471-488.
- W.Z. Kitto, A. Vince, and D.C. Wilson. An isomorphism between the $p$-adic integers and a ring associated with a tiling of $N$-space by permutohedra. Discrete Appl. Math. 52 (1994), no. 1, 39-51. MR1283243
- C.D. Lorenz, and R.M. Ziff. Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation. Journal of Physics A, 31 (1998) 8147-8157.
- A. Nahum, and J.T. Chalker. Universal statistics of vortex lines. Physical Review E, 85(3) (2012).
- A. Nahum, J.T. Chalker, P. Serna, M. Ortuno, and A.M. Somoza. 3d loop models and the cp^n-1 sigma model. Physical Review Letters, 107(11) (2011).
- R.J. Scherrer, and J.A. Frieman. Cosmic strings as random walks. Physical Review D, 33(12) (1986).
- O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221-288. MR1776084
- S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (2009), no. 1, 79-129. MR2494457
- J.H. Conway, and N.J.A. Sloane. Sphere packings, lattices and groups. Third edition. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer-Verlag, New York, 1999. lxxiv+703 pp. ISBN: 0-387-98585-9 MR1662447
- S. Smirnov. Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239-244. MR1851632
- S. Smirnov, W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (2001), no. 5-6, 729-744. MR1879816
- N. Sun. Conformally invariant scaling limits in planar critical percolation. Probab. Surv. 8 (2011), 155-209. MR2846901
- S. Tsarev. The geometry of a deformation of the standard addition on the integral lattice. ArXiv e-prints, January 2013.
- A. Vilenkin. Cosmic strings. The very early universe (Cambridge, 1982), 163--169, Cambridge Univ. Press, Cambridge, 1983. MR0746316
- G.M. Ziegler. Lecture on polytopes, volume 152. Springer Verlag, 1995.
This work is licensed under a Creative Commons Attribution 3.0 License.