Excited against the tide: a random walk with competing drifts

Mark Holmes

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

  • Volume: 48, Issue: 3, page 745-773
  • ISSN: 0246-0203

Abstract

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the k th cookie at a site determines the transition probabilities (to the left and right) for the k th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that the limiting velocity of the random walk is continuous in various parameters of the model and is monotone in the expected strength of the first cookie at the origin. We also give non-trivial examples where the first cookie drift is in the opposite direction to all subsequent cookie drifts and the velocity is zero. The proofs are based on a cut-times result of Bolthausen, Sznitman and Zeitouni, the lace expansion for self-interacting random walks of van der Hofstad and Holmes, and a coupling argument.

How to cite

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Holmes, Mark. "Excited against the tide: a random walk with competing drifts." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 745-773. <http://eudml.org/doc/272021>.

@article{Holmes2012,
abstract = {We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$th cookie at a site determines the transition probabilities (to the left and right) for the $k$th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that the limiting velocity of the random walk is continuous in various parameters of the model and is monotone in the expected strength of the first cookie at the origin. We also give non-trivial examples where the first cookie drift is in the opposite direction to all subsequent cookie drifts and the velocity is zero. The proofs are based on a cut-times result of Bolthausen, Sznitman and Zeitouni, the lace expansion for self-interacting random walks of van der Hofstad and Holmes, and a coupling argument.},
author = {Holmes, Mark},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {self-interacting random walk; cookie environment; lace expansion; monotonicity; self-interacting random walks},
language = {eng},
number = {3},
pages = {745-773},
publisher = {Gauthier-Villars},
title = {Excited against the tide: a random walk with competing drifts},
url = {http://eudml.org/doc/272021},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Holmes, Mark
TI - Excited against the tide: a random walk with competing drifts
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 745
EP - 773
AB - We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$th cookie at a site determines the transition probabilities (to the left and right) for the $k$th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that the limiting velocity of the random walk is continuous in various parameters of the model and is monotone in the expected strength of the first cookie at the origin. We also give non-trivial examples where the first cookie drift is in the opposite direction to all subsequent cookie drifts and the velocity is zero. The proofs are based on a cut-times result of Bolthausen, Sznitman and Zeitouni, the lace expansion for self-interacting random walks of van der Hofstad and Holmes, and a coupling argument.
LA - eng
KW - self-interacting random walk; cookie environment; lace expansion; monotonicity; self-interacting random walks
UR - http://eudml.org/doc/272021
ER -

References

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