Electronic Journal of Probability

Continuity of the time and isoperimetric constants in supercritical percolation

Olivier Garet, Régine Marchand, Eviatar B. Procaccia, and Marie Théret

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Abstract

We consider two different objects on supercritical Bernoulli percolation on the edges of $\mathbb{Z} ^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z} ^2$ is continuous in the percolation parameter. As a corollary we obtain that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdorff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z} ^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty ]$, such that $\mathbb{P} [t(e)<+\infty ] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property of the asymptotic shape previously proved by Cox and Kesten [8, 10, 20] for first-passage percolation with finite passage times.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 78, 35 pp.

Dates
Received: 23 December 2016
Accepted: 5 August 2017
First available in Project Euclid: 2 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1506931228

Digital Object Identifier
doi:10.1214/17-EJP90

Mathematical Reviews number (MathSciNet)
MR3710798

Zentralblatt MATH identifier
06797888

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

Keywords
continuity first-passage percolation time constant isoperimetric constant

Rights
Creative Commons Attribution 4.0 International License.

Citation

Garet, Olivier; Marchand, Régine; Procaccia, Eviatar B.; Théret, Marie. Continuity of the time and isoperimetric constants in supercritical percolation. Electron. J. Probab. 22 (2017), paper no. 78, 35 pp. doi:10.1214/17-EJP90. https://projecteuclid.org/euclid.ejp/1506931228


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References

  • [1] Peter Antal and Agoston Pisztora, On the chemical distance for supercritical Bernoulli percolation, Ann. Probab. 24 (1996), no. 2, 1036–1048.
  • [2] I. Benjamini and E. Mossel, On the mixing time of a simple random walk on the super critical percolation cluster, Probability theory and related fields 125 (2003), no. 3, 408–420.
  • [3] N. Berger, M. Biskup, C.E. Hoffman, and G. Kozma, Anomalous heat-kernel decay for random walk among bounded random conductances, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 44 (2008), no. 2, 374–392.
  • [4] Marek Biskup, Oren Louidor, Eviatar B. Procaccia, and Ron Rosenthal, Isoperimetry in two-dimensional percolation, Comm. Pure Appl. Math. 68 (2015), no. 9, 1483–1531.
  • [5] R. Cerf, The Wulff crystal in Ising and percolation models, Lecture Notes in Mathematics, vol. 1878, Springer-Verlag, Berlin, 2006, Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With a foreword by Jean Picard.
  • [6] Raphaël Cerf and Marie Théret, Weak shape theorem in first passage percolation with infinite passage times, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1351–1381.
  • [7] Olivier Couronné and Reda Jürg Messikh, Surface order large deviations for 2D FK-percolation and Potts models, Stochastic Process. Appl. 113 (2004), no. 1, 81–99.
  • [8] J. Theodore Cox, The time constant of first-passage percolation on the square lattice, Adv. in Appl. Probab. 12 (1980), no. 4, 864–879.
  • [9] J. Theodore Cox and Richard Durrett, Some limit theorems for percolation processes with necessary and sufficient conditions, Ann. Probab. 9 (1981), no. 4, 583–603.
  • [10] J. Theodore Cox and Harry Kesten, On the continuity of the time constant of first-passage percolation, J. Appl. Probab. 18 (1981), no. 4, 809–819.
  • [11] Luiz Fontes and Charles M. Newman, First passage percolation for random colorings of ${\textbf Z}^d$, Ann. Appl. Probab. 3 (1993), no. 3, 746–762.
  • [12] Olivier Garet and Régine Marchand, Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster, ESAIM Probab. Stat. 8 (2004), 169–199 (electronic).
  • [13] Olivier Garet and Régine Marchand, Large deviations for the chemical distance in supercritical Bernoulli percolation, Ann. Probab. 35 (2007), no. 3, 833–866.
  • [14] Olivier Garet and Régine Marchand, Moderate deviations for the chemical distance in Bernoulli percolation, ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010), 171–191.
  • [15] Julian Gold, Isoperimetry in supercritical bond percolation in dimensions three and higher, Available from arXiv:1602.05598, 2016.
  • [16] Geoffrey Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999.
  • [17] J. M. Hammersley and D. J. A. Welsh, First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif., Springer-Verlag, New York, 1965, pp. 61–110.
  • [18] H. Kesten, Aspects of first passage percolation, Lecture Notes in Math 1180 (1986), 125–264.
  • [19] Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982.
  • [20] Harry Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 125–264.
  • [21] Harry Kesten, Surfaces with minimal random weights and maximal flows: a higher dimensional version of first-passage percolation, Illinois Journal of Mathematics 31 (1987), no. 1, 99–166.
  • [22] T.M. Liggett, R.H. Schonmann, and A.M. Stacey, Domination by product measures, Ann. Probab. 25 (1997), 71–95.
  • [23] P. Mathieu and E. Remy, Isoperimetry and heat kernel decay on percolation clusters, The Annals of Probability 32 (2004), no. 1A, 100–128.
  • [24] G. Pete, A note on percolation on $\mathbb{Z} ^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), no. 37, 377–392.
  • [25] Agoston Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (1996), no. 4, 427–466.
  • [26] E.B. Procaccia and R. Rosenthal, Concentration estimates for the isoperimetric constant of the supercritical percolation cluster, Electron. Commun. Probab. 17 (2012), no. 30, 1–11.
  • [27] Ádám Timár, Boundary-connectivity via graph theory, Proc. Amer. Math. Soc. 141 (2013), no. 2, 475–480.
  • [28] Georg Wulff, Xxv. zur frage der geschwindigkeit des wachsthums und der auflösung der krystallflächen, Zeitschrift für Kristallographie-Crystalline Materials 34 (1901), no. 1, 449–530.