Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Zero–one law for directional transience of one dimensional excited random walks

Gideon Amir, Noam Berger, and Tal Orenshtein

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The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner (Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013) 105–157). As an application, a law of large numbers holds in these conditions.


La probabilité qu’une marche aléatoire unidimensionnelle excitée dans un environnement ergodique et elliptique soit transiente à gauche (à droite) est soit nulle soit un. Ceci résout un problème posé par Kosygina et Zerner (Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013) 105–157). Comme application, une loi des grands nombres est démontrée sous ces conditions.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 47-57.

Received: 23 May 2013
Revised: 20 February 2014
Accepted: 17 March 2014
First available in Project Euclid: 6 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Excited random walk Cookie walk Recurrence Directional transience Zero–one law Law of large numbers Limit theorem Random environment


Amir, Gideon; Berger, Noam; Orenshtein, Tal. Zero–one law for directional transience of one dimensional excited random walks. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 47--57. doi:10.1214/14-AIHP615.

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