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

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.

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.

DOI : https://doi.org/10.1214/10-AIHP403
Classification:  60K35,  82B20
Keywords: percolation, Lipschitz embedding, random surface, branching process, total progeny
@article{AIHPB_2012__48_2_309_0,
     author = {Grimmett, G. R. and Holroyd, A. E.},
     title = {Geometry of Lipschitz percolation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {2},
     year = {2012},
     pages = {309-326},
     doi = {10.1214/10-AIHP403},
     zbl = {1255.60167},
     mrnumber = {2954256},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_2_309_0}
}
Geometry of Lipschitz percolation. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, pp. 309-326. doi : 10.1214/10-AIHP403. http://www.numdam.org/item/AIHPB_2012__48_2_309_0/

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