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.
Mots clés : 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}, pages = {309--326}, publisher = {Gauthier-Villars}, volume = {48}, number = {2}, year = {2012}, doi = {10.1214/10-AIHP403}, mrnumber = {2954256}, zbl = {1255.60167}, language = {en}, url = {http://www.numdam.org/articles/10.1214/10-AIHP403/} }
TY - JOUR AU - Grimmett, G. R. AU - Holroyd, A. E. TI - Geometry of Lipschitz percolation JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 309 EP - 326 VL - 48 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP403/ DO - 10.1214/10-AIHP403 LA - en ID - AIHPB_2012__48_2_309_0 ER -
%0 Journal Article %A Grimmett, G. R. %A Holroyd, A. E. %T Geometry of Lipschitz percolation %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 309-326 %V 48 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP403/ %R 10.1214/10-AIHP403 %G en %F AIHPB_2012__48_2_309_0
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/
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