no. 2
Geometry of Lipschitz percolation
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 309-326.

Nous démontrons plusieurs résultats concernant la percolation Lipschitzienne. La probabilité critique pL pour l'existence d'une surface Lipschitzienne ouverte dans la percolation par site sur ℤd (lorsque d ≥ 2) satisfait l'estimation améliorée pL ≤ 1 - 1/[8(d - 1)]. Pour tout p > pL, la hauteur de la plus basse surface Lipschitzienne au-dessus de l'origine a une queue qui décroît exponentiellement vite. Lorsque p est suffisamment proche de 1, la taille des régions connexes de ℤd-1 au-dessus desquelles cette surface a une hauteur supérieure ou égale à 2 possède un comportement exponentiel étiré. Ce dernier résultat provient d'une inégalité stochastique qui montre que la plus basse surface est dominée stochastiquement par la frontière de l'union de certains ensembles aléatoires de ℤd indépendants et identiquement distribués.

We prove several facts concerning Lipschitz percolation, including the following. The critical probability pL for the existence of an open Lipschitz surface in site percolation on ℤd with d ≥ 2 satisfies the improved bound pL ≤ 1 - 1/[8(d - 1)]. Whenever p > pL, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. For p sufficiently close to 1, the connected regions of ℤd-1 above which the surface has height 2 or more exhibit stretched-exponential tail behaviour. The last statement is proved via a stochastic inequality stating that the lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of ℤd.

DOI : 10.1214/10-AIHP403
Classification : 60K35, 82B20
Mots clés : percolation, Lipschitz embedding, random surface, branching process, total progeny 
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     title = {Geometry of {Lipschitz} percolation},
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Grimmett, G. R.; Holroyd, A. E. Geometry of Lipschitz percolation. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 309-326. doi : 10.1214/10-AIHP403. http://www.numdam.org/articles/10.1214/10-AIHP403/

[1] P. Balister, B. Bollobás and A. Stacey. Improved upper bounds for the critical probability of oriented percolation in two dimensions. Random Structures Algorithms 5 (1994) 573-589. | MR | Zbl

[2] A. Baltrūnas, D. J. Daley and C. Klüppelberg. Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times. Stochastic Process. Appl. 111 (2004) 237-258. | MR | Zbl

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

[4] A. A. Borovkov and K. A. Borovkov. Asymptotic Analysis of Random Walks. Cambridge Univ. Press, Cambridge, 2008. | Zbl

[5] D. Denisov, A. B. Dieker and V. Shneer. Large deviations for random walks under subexponentiality: The big jump domain. Ann. Probab. 36 (2008) 1946-1991. | MR | Zbl

[6] N. Dirr, P. W. Dondl, G. R. Grimmett, A. E. Holroyd and M. Scheutzow. Lipschitz percolation. Electron. Comm. Probab. 15 (2010) 14-21. | MR | Zbl

[7] N. Dirr, P. W. Dondl and M. Scheutzow. Pinning of interfaces in random media. Preprint, 2009. Available at arXiv:0911.4254. | MR | Zbl

[8] R. L. Dobrushin. Gibbs state describing coexistence of phases for a three-dimensional Ising model. Theory Probab. Appl. 18 (1972) 582-600. | Zbl

[9] G. Gielis and G. R. Grimmett. Rigidity of the interface in percolation and random-cluster models. J. Statist. Phys. 109 (2002) 1-37. | MR | Zbl

[10] J.-B. Gouéré. Existence of subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36 (2008) 1209-1220. | MR | Zbl

[11] G. R. Grimmett. Percolation, 2nd edition. Springer, Berlin, 1999. | MR

[12] G. R. Grimmett. The Random-Cluster Model. Springer, Berlin, 2006. | MR | Zbl

[13] G. R. Grimmett and P. Hiemer. Directed percolation and random walk. In In and Out of Equilibrium 273-297. V. Sidoravicius (Ed.). Birkhäuser, Boston, 2002. | MR | Zbl

[14] G. R. Grimmett and A. E. Holroyd. Lattice embeddings in percolation. Ann. Probab. 40 (2012) 146-161. | MR | Zbl

[15] G. R. Grimmett and A. E. Holroyd. Plaquettes, spheres, and entanglement. Electron. J. Probab. 15 (2010) 1415-1428. | MR | Zbl

[16] C. C. Heyde. Two probability theorems and their applications to some first passage problems. J. Austral. Math. Soc. 4 (1964) 214-222. | MR | Zbl

[17] T. M. Liggett. Survival of discrete time growth models, with applications to oriented percolation. Ann. Appl. Prob. 5 (1995) 613-636. | MR | Zbl

[18] R. Meester and R. Roy. Continuum Percolation. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[19] C. E. M. Pearce and F. K. Fletcher. Oriented site percolation, phase transitions and probability bounds. J. Inequal. Pure Appl. Math. 6 (2005) Article 135. | MR | Zbl

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