Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment

Ross G. Pinsky

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Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site x, the probability of jumping to the right is ω(x)∈[½, 1), until the first time the process jumps to the left from site x, from which time onward the probability of jumping to the right is ½. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments {ω(x)}xZ. In deterministic environments, we also study the speed of the process.


Considérons une variante de la marche aléatoire simple et symétrique sur les entiers, avec le mécanisme de transition suivant: A chaque site x, la probabilité de sauter à droite est ω(x)∈[½, 1), jusqu’à la première fois que le processus saute à gauche du site x, après laquelle la probabilité de sauter à droite est ½. Nous examinons les propriétés de transience/recurrence pour ce processus, dans les environnements déterministes et aussi dans les environnements stationnaires et ergodiques {ω(x)}xZ. Dans les environnements déterministes, nous étudions aussi la vitesse du processus.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 949-964.

First available in Project Euclid: 4 November 2010

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments

Excited random walk Cookies Transience Recurrence Ballistic


Pinsky, Ross G. Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 949--964. doi:10.1214/09-AIHP331.

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