Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Geometry of Lipschitz percolation

G. R. Grimmett and A. E. Holroyd

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 309-326.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148200

Digital Object Identifier
doi:10.1214/10-AIHP403

Mathematical Reviews number (MathSciNet)
MR2954256

Zentralblatt MATH identifier
1255.60167

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Percolation Lipschitz embedding Random surface Branching process Total progeny

Citation

Grimmett, G. R.; Holroyd, A. E. Geometry of Lipschitz percolation. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 309--326. doi:10.1214/10-AIHP403. https://projecteuclid.org/euclid.aihp/1334148200


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