Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster
Olivier Garet; Régine Marchand
ESAIM: Probability and Statistics (2010)
- Volume: 8, page 169-199
- ISSN: 1292-8100
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topGaret, Olivier, and Marchand, Régine. "Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster." ESAIM: Probability and Statistics 8 (2010): 169-199. <http://eudml.org/doc/104317>.
@article{Garet2010,
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.; random environment},
language = {eng},
month = {3},
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/104317},
volume = {8},
year = {2010},
}
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
DA - 2010/3//
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.; random environment
UR - http://eudml.org/doc/104317
ER -
References
top- 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.
- P. Antal and A. Pisztora, On the chemical distance for supercritical Bernoulli percolation. Ann. Probab.24 (1996) 1036–1048.
- D. Boivin, First passage percolation: the stationary case. Probab. Theory Related Fields86 (1990) 491–499.
- J.R. Brown, Ergodic theory and topological dynamics. Academic Press, Harcourt Brace Jovanovich Publishers, New York. Pure Appl. Math.70 (1976).
- R.M. Burton and M. Keane, Density and uniqueness in percolation. Comm. Math. Phys.121 (1989) 501–505.
- J.T. Cox, The time constant of first-passage percolation on the square lattice. Adv. Appl. Probab.12 (1980) 864–879.
- J.T. Cox and R. Durrett, Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab.9 (1981) 583–603.
- J.T. Cox and H. Kesten, On the continuity of the time constant of first-passage percolation. J. Appl. Probab.18 (1981) 809–819.
- R. Durrett and T.M. Liggett, The shape of the limit set in Richardson's growth model. Ann. Probab.9 (1981) 186–193.
- O. Garet, Percolation transition for some excursion sets. Electron. J. Probab.9 (2004) 255–292 (electronic).
- O. Häggström and R. Meester, Asymptotic shapes for stationary first passage percolation. Ann. Probab.23 (1995) 1511–1522.
- 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.
- 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.
- H. Kesten and Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab.18 (1990) 537–555.
- R. Marchand, Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab.12 (2002) 1001–1038.
- D. Richardson, Random growth in a tessellation. Proc. Cambridge Philos. Soc.74 (1973) 515–528.
- Y.G. Sinai, Introduction to ergodic theory. Princeton University Press, Princeton, N.J., Translated by V. Scheffer. Math. Notes18 (1976).
- W.F. Stout, Almost sure convergence. Academic Press, A subsidiary of Harcourt Brace Jovanovich, Publishers, New York-London. Probab. Math. Statist.24 (1974).
- J. van den Berg and H. Kesten, Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab.3 (1993) 56–80.
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