The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on 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.
Keywords: percolation, first-passage percolation, chemical distance, infinite cluster, asymptotic shape, random environment
@article{PS_2004__8__169_0,
author = {Garet, Olivier and Marchand, R\'egine},
title = {Asymptotic shape for the chemical distance and first-passage percolation on the infinite {Bernoulli} cluster},
journal = {ESAIM: Probability and Statistics},
pages = {169--199},
year = {2004},
publisher = {EDP Sciences},
volume = {8},
doi = {10.1051/ps:2004009},
mrnumber = {2085613},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps:2004009/}
}
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 SP - 169 EP - 199 VL - 8 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps:2004009/ DO - 10.1051/ps:2004009 LA - en ID - PS_2004__8__169_0 ER -
%0 Journal Article %A Garet, Olivier %A Marchand, Régine %T Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster %J ESAIM: Probability and Statistics %D 2004 %P 169-199 %V 8 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps:2004009/ %R 10.1051/ps:2004009 %G en %F PS_2004__8__169_0
Garet, Olivier; Marchand, Régine. Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 169-199. doi: 10.1051/ps:2004009
[1] , and, Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 (1987) 505-531. | Zbl
[2] and, On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036-1048. | Zbl
[3] , First passage percolation: the stationary case. Probab. Theory Related Fields 86 (1990) 491-499. | Zbl
[4] , Ergodic theory and topological dynamics. Academic Press, Harcourt Brace Jovanovich Publishers, New York. Pure Appl. Math. 70 (1976). | Zbl | MR
[5] and, Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989) 501-505. | Zbl
[6] , The time constant of first-passage percolation on the square lattice. Adv. Appl. Probab. 12 (1980) 864-879. | Zbl
[7] and, Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 (1981) 583-603. | Zbl
[8] and, On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18 (1981) 809-819. | Zbl
[9] and, The shape of the limit set in Richardson's growth model. Ann. Probab. 9 (1981) 186-193. | Zbl
[10] , Percolation transition for some excursion sets. Electron. J. Probab. 9 (2004) 255-292 (electronic). | Zbl
[11] and, Asymptotic shapes for stationary first passage percolation. Ann. Probab. 23 (1995) 1511-1522. | Zbl
[12] and, 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. | Zbl
[13] , 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. | Zbl
[14] and, The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990) 537-555. | Zbl
[15] , Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 (2002) 1001-1038. | Zbl
[16] , Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 (1973) 515-528. | Zbl
[17] , Introduction to ergodic theory. Princeton University Press, Princeton, N.J., Translated by V. Scheffer. Math. Notes 18 (1976). | Zbl | MR
[18] , Almost sure convergence. Academic Press, A subsidiary of Harcourt Brace Jovanovich, Publishers, New York-London. Probab. Math. Statist. 24 (1974). | Zbl | MR
[19] and, Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3 (1993) 56-80. | Zbl
Cité par Sources :
