Electronic Journal of Probability

The shape of large balls in highly supercritical percolation

Anne-Laure Basdevant, Nathanaël Enriquez, Lucas Gerin, and Jean-Baptiste Gouéré

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Abstract

We exploit a connection between distances in the infinite percolation cluster, when the parameter is close to one, and the discrete-time TASEP on Z. This shows that when the parameter goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.

Article information

Source
Electron. J. Probab. Volume 19 (2014), paper no. 26, 14 pp.

Dates
Accepted: 28 February 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065668

Digital Object Identifier
doi:10.1214/EJP.v19-3062

Mathematical Reviews number (MathSciNet)
MR3174838

Zentralblatt MATH identifier
1294.60118

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
first-passage percolation supercritical percolation TASEP

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Basdevant, Anne-Laure; Enriquez, Nathanaël; Gerin, Lucas; Gouéré, Jean-Baptiste. The shape of large balls in highly supercritical percolation. Electron. J. Probab. 19 (2014), paper no. 26, 14 pp. doi:10.1214/EJP.v19-3062. https://projecteuclid.org/euclid.ejp/1465065668


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References

  • Auffinger, Antonio; Daon, Michael. Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Related Fields 156 (2013), no. 1-2, 193–227.
  • Basdevant, Anne-Laure; Enriquez, Nathanaël; Gerin, Lucas. Distances in the highly supercritical percolation cluster. Ann. Probab. 41 (2013), no. 6, 4342–4358.
  • N.D. Blair-Stahn. First passage percolation and competition models (2010). arXiv:1005.0649.
  • Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999–1040.
  • Garet, Olivier; Marchand, Régine. Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat. 8 (2004), 169–199 (electronic).
  • 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
  • Hammersley, J. M.; Welsh, D. J. A. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. 1965 Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. pp. 61–110 Springer-Verlag, New York
  • W.Jockusch, J.Propp, P.Shor. Random domino tilings and the arctic circle theorem (1995). arXiv:math/9801068.
  • Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
  • Kriecherbauer, Thomas; Krug, Joachim. A pedestrian's view on interacting particle systems, KPZ universality and random matrices. J. Phys. A 43 (2010), no. 40, 403001, 41 pp.
  • Marchand, R. Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 (2002), no. 3, 1001–1038.
  • T.Seppäläinen. Lecture Notes on the Corner Growth Model (2008). Available at http://www.math.wisc.edu/~seppalai/.