## Electronic Journal of Probability

### Excited deterministic walk in a random environment

#### Abstract

Excited deterministic walk in a random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $\mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site $x$ arbitrarily many times before getting stuck in a loop, which may or may not contain $x$. Nevertheless, we establish monotonicity results on the environment that imply large deviations.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 44, 19 pp.

Dates
Accepted: 15 April 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067150

Digital Object Identifier
doi:10.1214/EJP.v20-3874

Mathematical Reviews number (MathSciNet)
MR3339864

Zentralblatt MATH identifier
1321.60215

Subjects
Primary: 60F10: Large deviations

Rights

#### Citation

Matic, Ivan; Sivakoff, David. Excited deterministic walk in a random environment. Electron. J. Probab. 20 (2015), paper no. 44, 19 pp. doi:10.1214/EJP.v20-3874. https://projecteuclid.org/euclid.ejp/1465067150

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