Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Excited random walk with periodic cookies

Gady Kozma, Tal Orenshtein, and Igor Shinkar

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In this paper we consider an excited random walk (ERW) on $\mathbb{Z}$ in identically piled periodic environment. This is a discrete time process on $\mathbb{Z}$ defined by parameters $(p_{1},\dots,p_{M})\in[0,1]^{M}$ for some positive integer $M$, where the walker upon the $i$th visit to $z\in\mathbb{Z}$ moves to $z+1$ with probability $p_{i\ (\mathrm{mod}\ M)}$, and moves to $z-1$ with probability $1-p_{i\ (\mathrm{mod}\ M)}$. We give an explicit formula in terms of the parameters $(p_{1},\dots,p_{M})$ which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that $\frac{1}{M}\sum_{i=1}^{M}p_{i}=\frac{1}{2}$ all behaviors are possible, and may depend on the order of the $p_{i}$. Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.


Dans ce papier, nous considérons une marche aléatoire excitée (MAE) sur $\mathbb{Z}$ en environnement empilé de manière identique et périodique. Il s’agit d’un processus à temps discret sur $\mathbb{Z}$ défini par des paramètres $(p_{1},\dots,p_{M})\in[0,1]^{M}$ pour un certain entier strictement positif $M$, où le marcheur, après la $i$ième visite au site $z\in\mathbb{Z}$ se déplace soit en $z+1$ avec probabilité $p_{i\ (\mathrm{mod}\ M)}$, soit en $z-1$ avec probabilité $1-p_{i\ (\mathrm{mod}\ M)}$. Nous donnons une formule explicite en fonction des paramètres $(p_{1},\dots,p_{M})$ qui détermine si la marche est récurrente, transiente vers la gauche, ou transiente vers la droite. En particulier, dans le cas où $\frac{1}{M}\sum_{i=1}^{M}p_{i}=\frac{1}{2}$, tous les comportements sont possibles et peuvent dépendre de l’ordre des $p_{i}$. Notre approche permet de retrouver directement certains résultats connus sur les MAE et les processus de branchement sans immigration.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1023-1049.

Received: 2 December 2013
Revised: 20 December 2014
Accepted: 29 January 2015
First available in Project Euclid: 28 July 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] 60J85: Applications of branching processes [See also 92Dxx]

Excited random walk Cookie walk Recurrence Transience Bessel process Lyapunov function Branching process with migration


Kozma, Gady; Orenshtein, Tal; Shinkar, Igor. Excited random walk with periodic cookies. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1023--1049. doi:10.1214/15-AIHP669.

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