Electronic Journal of Probability

Existence and continuity of the flow constant in first passage percolation

Raphaël Rossignol and Marie Théret

Full-text: Open access

Abstract

We consider the model of i.i.d. first passage percolation on $\mathbb{Z} ^d$, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution $G$ on $ [0,+\infty ]$ (including $+\infty $). Whereas the time constant is associated to the study of $1$-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of $(d-1)$-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that $G(\{+\infty \} ) < p_c(d)$ (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution $G$.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 99, 42 pp.

Dates
Received: 13 September 2017
Accepted: 20 August 2018
First available in Project Euclid: 26 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1537927580

Digital Object Identifier
doi:10.1214/18-EJP214

Mathematical Reviews number (MathSciNet)
MR3862614

Zentralblatt MATH identifier
06964793

Subjects
Primary: 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 maximal flow minimal cutset continuity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Rossignol, Raphaël; Théret, Marie. Existence and continuity of the flow constant in first passage percolation. Electron. J. Probab. 23 (2018), paper no. 99, 42 pp. doi:10.1214/18-EJP214. https://projecteuclid.org/euclid.ejp/1537927580


Export citation

References

  • [1] M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, Journal für die Reine und Angewandte Mathematik 323 (1981), 53–67.
  • [2] Antonio Auffinger, Michael Damron, and Jack Hanson, 50 years of first passage percolation, University Lecture Series, vol. 68, American Mathematical Society, 2017.
  • [3] Béla Bollobás, Graph theory, Graduate Texts in Mathematics, vol. 63, Springer-Verlag, New York, 1979, An introductory course.
  • [4] Raphaël Cerf, The Wulff crystal in Ising and percolation models, École d’Été de Probabilités de Saint Flour, Lecture Notes in Mathematics, no. 1878, Springer-Verlag, 2006.
  • [5] Raphaël Cerf and Marie Théret, Law of large numbers for the maximal flow through a domain of $\mathbb{R} ^d$ in first passage percolation, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3665–3702.
  • [6] Raphaël Cerf and Marie Théret, Lower large deviations for the maximal flow through a domain of $\mathbb{R} ^d$ in first passage percolation, Probability Theory and Related Fields 150 (2011), 635–661.
  • [7] Raphaël Cerf and Marie Théret, Upper large deviations for the maximal flow through a domain of $\mathbb{R} ^d$ in first passage percolation, Annals of Applied Probability 21 (2011), no. 6, 2075–2108.
  • [8] Raphaël Cerf and Marie Théret, Maximal stream and minimal cutset for first passage percolation through a domain of $\mathbb{R} ^d$, Ann. Probab. 42 (2014), no. 3, 1054–1120.
  • [9] 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.
  • [10] J. Theodore Cox, The time constant of first-passage percolation on the square lattice, Adv. in Appl. Probab. 12 (1980), no. 4, 864–879.
  • [11] 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.
  • [12] 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.
  • [13] Luiz Fontes and Charles M. Newman, First passage percolation for random colorings of $\mathbf{Z} ^d$, Ann. Appl. Probab. 3 (1993), no. 3, 746–762.
  • [14] 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).
  • [15] Olivier Garet, Régine Marchand, Eviatar B. Procaccia, and Marie Théret, Continuity of the time and isoperimetric constants in supercritical percolation, Electron. J. Probab. 22 (2017), Paper No. 78, 35.
  • [16] G. Grimmett and H. Kesten, First-passage percolation, network flows and electrical resistances, Z. Wahrsch. Verw. Gebiete 66 (1984), no. 3, 335–366.
  • [17] Geoffrey Grimmett, Percolation, Springer-Verlag, 1989.
  • [18] A. Gut, Complete convergence for arrays, Period. Math. Hungar. 25 (1992), no. 1, 51–75.
  • [19] Allan Gut, On complete convergence in the law of large numbers for subsequences, Ann. Probab. 13 (1985), no. 4, 1286–1291.
  • [20] 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.
  • [21] Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser Boston, Mass., 1982.
  • [22] 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.
  • [23] 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.
  • [24] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499–510.
  • [25] U. Krengel and R. Pyke, Uniform pointwise ergodic theorems for classes of averaging sets and multiparameter subadditive processes, Stochastic Process. Appl. 26 (1987), no. 2, 289–296.
  • [26] Raphaël Rossignol and Marie Théret, Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation, Stochastic Processes and their Applications 120 (2010), 873–900.
  • [27] Raphaël Rossignol and Marie Théret, Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 4, 1093–1131.
  • [28] R. T. Smythe, Multiparameter subadditive processes, Ann. Probability 4 (1976), no. 5, 772–782.
  • [29] Yu Zhang, Critical behavior for maximal flows on the cubic lattice, Journal of Statistical Physics 98 (2000), no. 3-4, 799–811.
  • [30] Yu Zhang, Limit theorems for maximum flows on a lattice, Probability Theory and Related Fields (2018), no. 1-2, 149–202.