no. 3
Seven-dimensional forest fires
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 862-866.

Cette article montre que dans la percolation de Bernoulli par arête en grande dimension, retirer d’une composante connexe infinie de faible densité une composante connexe de densité beaucoup plus faible laisse une composante connexe infinie. Cette observation a des implications pour le processus de feux de forêt de van den Berg–Brouwer, également connu sous le nom de percolation auto-destructive, en dimension suffisamment grande.

We show that in high dimensional Bernoulli bond percolation, removing from a thin infinite cluster a much thinner infinite cluster leaves an infinite component. This observation has implications for the van den Berg–Brouwer forest fire process, also known as self-destructive percolation, for dimension high enough.

DOI : 10.1214/13-AIHP587
Mots clés : near-critical percolation, static renormalization 
@article{AIHPB_2015__51_3_862_0,
     author = {Ahlberg, Daniel and Duminil-Copin, Hugo and Kozma, Gady and Sidoravicius, Vladas},
     title = {Seven-dimensional forest fires},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {862--866},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/13-AIHP587},
     mrnumber = {3365964},
     zbl = {1323.60123},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/13-AIHP587/}
}
TY  - JOUR
AU  - Ahlberg, Daniel
AU  - Duminil-Copin, Hugo
AU  - Kozma, Gady
AU  - Sidoravicius, Vladas
TI  - Seven-dimensional forest fires
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 862
EP  - 866
VL  - 51
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/13-AIHP587/
DO  - 10.1214/13-AIHP587
LA  - en
ID  - AIHPB_2015__51_3_862_0
ER  - 
%0 Journal Article
%A Ahlberg, Daniel
%A Duminil-Copin, Hugo
%A Kozma, Gady
%A Sidoravicius, Vladas
%T Seven-dimensional forest fires
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 862-866
%V 51
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/13-AIHP587/
%R 10.1214/13-AIHP587
%G en
%F AIHPB_2015__51_3_862_0
Ahlberg, Daniel; Duminil-Copin, Hugo; Kozma, Gady; Sidoravicius, Vladas. Seven-dimensional forest fires. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 862-866. doi : 10.1214/13-AIHP587. http://www.numdam.org/articles/10.1214/13-AIHP587/

[1] A. Aharony and D. Stauffer. Introduction to Percolation Theory, 2nd edition. Taylor and Francis, London, 1994. | MR | Zbl

[2] D. Ahlberg, V. Sidoravicius and J. Tykesson. Bernoulli and self-destructive percolation on non-amenable graphs. Electron. J. Probab. 19 (2014) 1–6. | MR | Zbl

[3] N. Berger, M. Biskup, C. E. Hoffman and G. Kozma. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2) (2008) 374–392. | Numdam | MR | Zbl

[4] G. Grimmett, A. Holroyd and G. Kozma. Percolation of finite clusters and infinite surfaces. Math. Proc. Cambridge Phil. Soc. 156 (2014) 263–279. | DOI | MR | Zbl

[5] 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. | MR | Zbl

[6] T. Hara. Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 (2) (2008) 530–593. | MR | Zbl

[7] T. Hara, R. Van Der Hofstad and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 (1) (2003) 349–408. | MR | Zbl

[8] H. Kesten. Percolation Theory for Mathematicians. Progress in Probability and Statistics 2. Birkhäuser, Boston, MA, 1982. | DOI | MR | Zbl

[9] G. Kozma and A. Nachmias. Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 (2) (2011) 375–409. | MR | Zbl

[10] T. M. Liggett, R. H. Schonmann and A. M. Stacey. Domination by product measures. Ann. Probab. 25 (1) (1997) 71–95. | MR | Zbl

[11] J. Van Den Berg and R. Brouwer. Self-destructive percolation. Random Structures Algorithms 24 (4) (2004) 480–501. | MR | Zbl

[12] J. Van Den Berg, R. Brouwer and B. Vágvölgyi. Box-crossings and continuity results for self-destructive percolation in the plane. In In and Out of Equilibrium 2 117–135. Progr. Probab. 60. Birkhäuser, Basel, 2008. | MR | Zbl

[13] J. Van Den Berg and B. N. B. De Lima. Linear lower bounds for δ c (p) for a class of 2D self-destructive percolation models. Random Structures Algorithms 34 (4) (2009) 520–526. | MR | Zbl

[14] J. Van Den Berg and H. Kesten. Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 (3) (1985) 556–569. | MR | Zbl

Cité par Sources :