Journal article

Excited against the tide: A random walk with competing drifts

Mark Holmes

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | INST MATHEMATICAL STATISTICS | Published : 2012

Abstract

We study excited random walks in i.i.d. random cookie environments in high dimensions, where the kth cookie at a site determines the transition probabilities (to the left and right) for the kth 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 ..

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University of Melbourne Researchers

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Funding Acknowledgements

This work was supported in part by a FRDF grant from the University of Auckland. The author would like to thank Takashi Hara for providing the SRW Green's functions upper bounds, Martin Zerner for helpful discussions, and Duncan Temple Lang and Stephen Cope for assistance with the simulations. We also thank an anonymous referee for many helpful suggestions and for asking about possible extensions of the results (in a previous version of this work) that has given rise to the present version of the paper.