Scaling of a random walk on a supercritical contact process

F. den Hollander; R. S. dos Santos

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1276-1300
  • ISSN: 0246-0203

Abstract

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We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.

How to cite

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den Hollander, F., and dos Santos, R. S.. "Scaling of a random walk on a supercritical contact process." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1276-1300. <http://eudml.org/doc/272085>.

@article{denHollander2014,
abstract = {We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.},
author = {den Hollander, F., dos Santos, R. S.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; dynamic random environment; contact process; strong law of large numbers; functional central limit theorem; large deviation principle; Space-time cones; clusters of infections; coupling; regeneration times; supercritical contact process; space-time cones},
language = {eng},
number = {4},
pages = {1276-1300},
publisher = {Gauthier-Villars},
title = {Scaling of a random walk on a supercritical contact process},
url = {http://eudml.org/doc/272085},
volume = {50},
year = {2014},
}

TY - JOUR
AU - den Hollander, F.
AU - dos Santos, R. S.
TI - Scaling of a random walk on a supercritical contact process
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1276
EP - 1300
AB - We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.
LA - eng
KW - random walk; dynamic random environment; contact process; strong law of large numbers; functional central limit theorem; large deviation principle; Space-time cones; clusters of infections; coupling; regeneration times; supercritical contact process; space-time cones
UR - http://eudml.org/doc/272085
ER -

References

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