Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Weak shape theorem in first passage percolation with infinite passage times

Raphaël Cerf and Marie Théret

Full-text: Open access

Abstract

We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^{d}$: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty]>p_{c}(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]<p_{c}(d)$.

Résumé

Nous considérons le modèle standard de percolation de premier passage sur $\mathbb{Z}^{d}$ : nous associons à chaque arête $e$ du graphe un temps de passage $t(e)$ à valeurs dans $[0,+\infty]$, tel que $\mathbb{P}[t(e)<+\infty]>p_{c}(d)$. De façon équivalente, nous considérons un modèle de percolation de premier passage standard (fini) sur le graphe obtenu par une percolation de Bernoulli sur-critique réalisée indépendamment. Nous prouvons un théorème de forme faible sans aucune hypothèse de moment. Nous prouvons aussi que la constante de temps correspondante est strictement positive si et seulement si $\mathbb{P}[t(e)=0]<p_{c}(d)$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1351-1381.

Dates
Received: 21 November 2014
Revised: 10 April 2015
Accepted: 1 May 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723523

Digital Object Identifier
doi:10.1214/15-AIHP686

Mathematical Reviews number (MathSciNet)
MR3531712

Zentralblatt MATH identifier
1350.60100

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
First passage percolation Time constant Shape theorem

Citation

Cerf, Raphaël; Théret, Marie. Weak shape theorem in first passage percolation with infinite passage times. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1351--1381. doi:10.1214/15-AIHP686. https://projecteuclid.org/euclid.aihp/1469723523


Export citation

References

  • [1] D. Ahlberg. A Hsu–Robbins–Erdös strong law in first-passage percolation. Ann. Probab. 43 (2015) 1992–2025.
  • [2] E. D. Andjel, N. Chabot and E. Saada. A shape theorem for an epidemic model in dimension $d\geq3$, 2011. Available at http://arxiv.org/abs/1110.0801.
  • [3] P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (2) (1996) 1036–1048.
  • [4] I. Benjamini, G. Kalai and O. Schramm. First passage percolation has sublinear distance variance. Ann. Probab. 31 (4) (2003) 1970–1978.
  • [5] M. Björklund. The asymptotic shape theorem for generalized first passage percolation. Ann. Probab. 38 (2) (2010) 632–660.
  • [6] D. Boivin. First passage percolation: The stationary case. Probab. Theory Related Fields 86 (4) (1990) 491–499.
  • [7] S. Chatterjee and P. S. Dey. Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Related Fields 156 (3–4) (2013) 613–663.
  • [8] J. T. Chayes, L. Chayes, G. R. Grimmett, H. Kesten and R. H. Schonmann. The correlation length for the high-density phase of Bernoulli percolation. Ann. Probab. 17 (4) (1989) 1277–1302.
  • [9] J. T. Cox and R. Durrett. Limit theorems for the spread of epidemics and forest fires. Stochastic Process. Appl. 30 (2) (1988) 171–191.
  • [10] J. T. Cox. The time constant of first-passage percolation on the square lattice. Adv. in Appl. Probab. 12 (4) (1980) 864–879.
  • [11] J. T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 (4) (1981) 583–603.
  • [12] J. T. Cox and H. Kesten. On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18 (4) (1981) 809–819.
  • [13] R. Durrett and R. H. Schonmann. Large deviations for the contact process and two-dimensional percolation. Probab. Theory Related Fields 77 (4) (1988) 583–603.
  • [14] O. Garet and R. Marchand. Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat. 8 (2004) 169–199 (electronic).
  • [15] O. Garet and R. Marchand. Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35 (3) (2007) 833–866.
  • [16] O. Garet and R. Marchand. Moderate deviations for the chemical distance in Bernoulli percolation. ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 171–191.
  • [17] G. R. Grimmett and J. M. Marstrand. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 (1879) (1990) 439–457.
  • [18] G. Grimmett. Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer-Verlag, Berlin, 1999.
  • [19] J. M. Hammersley and D. J. A. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, CA 61–110. Springer-Verlag, New York, 1965.
  • [20] H. Kesten. Percolation Theory for Mathematicians. Progress in Probability and Statistics 2. Birkhäuser, Boston, MA, 1982.
  • [21] H. Kesten. Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour, XIV – 1984 125–264. Lecture Notes in Math. 1180. Springer, Berlin, 1986.
  • [22] J. Kingman. Subadditive ergodic theory. Ann. Probab. 1 (6) (1973) 883–899.
  • [23] U. Krengel. Ergodic Theorems. de Gruyter Studies in Mathematics 6. Walter de Gruyter & Co., Berlin, 1985.
  • [24] J.-C. Mourrat. Lyapunov exponents, shape theorems and large deviations for the random walk in random potential. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) 165–211.
  • [25] D. Richardson. Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 (1973) 515–528.
  • [26] J. Scholler. Modèles dépendants en percolation de premier passage. Ph.D. thesis, Université de Lorraine, 2013.
  • [27] Y. Zhang. A shape theorem for epidemics and forest fires with finite range interactions. Ann. Probab. 21 (4) (1993) 1755–1781.