Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Olivier Garet; Régine Marchand

ESAIM: Probability and Statistics (2004)

  • Volume: 8, page 169-199
  • ISSN: 1292-8100

Abstract

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The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given.

How to cite

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Garet, Olivier, and Marchand, Régine. "Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster." ESAIM: Probability and Statistics 8 (2004): 169-199. <http://eudml.org/doc/244937>.

@article{Garet2004,
abstract = {The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb \{Z\}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given.},
author = {Garet, Olivier, Marchand, Régine},
journal = {ESAIM: Probability and Statistics},
keywords = {percolation; first-passage percolation; chemical distance; infinite cluster; asymptotic shape; random environment; Percolation},
language = {eng},
pages = {169-199},
publisher = {EDP-Sciences},
title = {Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster},
url = {http://eudml.org/doc/244937},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Garet, Olivier
AU - Marchand, Régine
TI - Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster
JO - ESAIM: Probability and Statistics
PY - 2004
PB - EDP-Sciences
VL - 8
SP - 169
EP - 199
AB - The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb {Z}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given.
LA - eng
KW - percolation; first-passage percolation; chemical distance; infinite cluster; asymptotic shape; random environment; Percolation
UR - http://eudml.org/doc/244937
ER -

References

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  1. [1] M. Aizenman, H. Kesten and C.M. Newman, Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 (1987) 505–531. Zbl0642.60102
  2. [2] P. Antal and A. Pisztora, On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036–1048. Zbl0871.60089
  3. [3] D. Boivin, First passage percolation: the stationary case. Probab. Theory Related Fields 86 (1990) 491–499. Zbl0685.60103
  4. [4] J.R. Brown, Ergodic theory and topological dynamics. Academic Press, Harcourt Brace Jovanovich Publishers, New York. Pure Appl. Math. 70 (1976). Zbl0334.28011MR492177
  5. [5] R.M. Burton and M. Keane, Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989) 501–505. Zbl0662.60113
  6. [6] J.T. Cox, The time constant of first-passage percolation on the square lattice. Adv. Appl. Probab. 12 (1980) 864–879. Zbl0442.60096
  7. [7] J.T. Cox and R. Durrett, Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 (1981) 583–603. Zbl0462.60012
  8. [8] J.T. Cox and H. Kesten, On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18 (1981) 809–819. Zbl0474.60085
  9. [9] R. Durrett and T.M. Liggett, The shape of the limit set in Richardson’s growth model. Ann. Probab. 9 (1981) 186–193. Zbl0457.60083
  10. [10] O. Garet, Percolation transition for some excursion sets. Electron. J. Probab. 9 (2004) 255–292 (electronic). Zbl1065.60147
  11. [11] O. Häggström and R. Meester, Asymptotic shapes for stationary first passage percolation. Ann. Probab. 23 (1995) 1511–1522. Zbl0852.60104
  12. [12] 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, Calif., Springer-Verlag, New York (1965) 61–110. Zbl0143.40402
  13. [13] H. Kesten, Aspects of first passage percolation, in École d’été de probabilités de Saint-Flour, XIV–1984, Springer, Berlin. Lect. Notes Math. 1180 (1986) 125–264. Zbl0602.60098
  14. [14] H. Kesten and Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990) 537–555. Zbl0705.60092
  15. [15] R. Marchand, Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 (2002) 1001–1038. Zbl1062.60100
  16. [16] D. Richardson, Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 (1973) 515–528. Zbl0295.62094
  17. [17] Y.G. Sinai, Introduction to ergodic theory. Princeton University Press, Princeton, N.J., Translated by V. Scheffer. Math. Notes 18 (1976). Zbl0375.28011MR584788
  18. [18] W.F. Stout, Almost sure convergence. Academic Press, A subsidiary of Harcourt Brace Jovanovich, Publishers, New York-London. Probab. Math. Statist. 24 (1974). Zbl0321.60022MR455094
  19. [19] J. van den Berg and H. Kesten, Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3 (1993) 56–80. Zbl0771.60092

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