Open Access
2015 Excited deterministic walk in a random environment
Ivan Matic, David Sivakoff
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Electron. J. Probab. 20: 1-19 (2015). DOI: 10.1214/EJP.v20-3874


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.


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Ivan Matic. David Sivakoff. "Excited deterministic walk in a random environment." Electron. J. Probab. 20 1 - 19, 2015.


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

zbMATH: 1321.60215
MathSciNet: MR3339864
Digital Object Identifier: 10.1214/EJP.v20-3874

Primary: 60F10

Keywords: deterministic walks , excited random environments , large deviations

Vol.20 • 2015
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