Geometry of Lipschitz percolation

G. R. Grimmett; A. E. Holroyd

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 309-326
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Grimmett, G. R., and Holroyd, A. E.. "Geometry of Lipschitz percolation." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 309-326. <http://eudml.org/doc/271966>.

@article{Grimmett2012,
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 &gt; 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.},
author = {Grimmett, G. R., Holroyd, A. E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {percolation; Lipschitz embedding; random surface; branching process; total progeny; branching processes},
language = {eng},
number = {2},
pages = {309-326},
publisher = {Gauthier-Villars},
title = {Geometry of Lipschitz percolation},
url = {http://eudml.org/doc/271966},
volume = {48},
year = {2012},
}

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
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 309
EP - 326
AB - 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 &gt; 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.
LA - eng
KW - percolation; Lipschitz embedding; random surface; branching process; total progeny; branching processes
UR - http://eudml.org/doc/271966
ER -

References

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