Excited against the tide: a random walk with competing drifts

Holmes, Mark

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 745-773.

Nous étudions des marches aléatoires excitées dans un environnement de cookies indépendants en grande dimension, où le $k$ième cookie d’un site détermine le taux de transition (vers la droite ou la gauche) pour le $k$ième départ de ce site. Nous montrons qu’en grande dimension, quand le taux de saut moyen vers la droite du premier cookie est suffisamment grand, la vitesse est strictement positive, quelque soit l’amplitude et le signe des cookies suivants. Sous des conditions supplémentaires sur l’environnement des cookies, nous montrons que la vitesse est une fonction continue des divers paramètres du modèle et est monotone en la force moyenne du cookie à l’origine. Nous donnons aussi des examples non-triviaux où la dérive du premier cookie est dans le sens opposé à toutes les autres et où la vitesse est nulle. Les preuves se basent sur un résultat de temps de coupure de Bolthausen, Sznitman et Zeitouni, le développement en lacets de marches aléatoires auto-interagissantes de van der Hofstad et Holmes, et un argument de couplage.

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.

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