## Electronic Journal of Probability

### Random walk on random walks

#### Abstract

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$. At each step the random walk performs a nearest-neighbour jump, moving to the right with probability $p_{\circ}$ when it is on a vacant site and probability $p_{\bullet}$ when it is on an occupied site. Assuming that $p_\circ \in (0,1)$ and $p_\bullet \neq \tfrac12$, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided $\rho$ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 95, 35 pp.

Dates
Accepted: 12 September 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067201

Digital Object Identifier
doi:10.1214/EJP.v20-4437

Mathematical Reviews number (MathSciNet)
MR3399831

Zentralblatt MATH identifier
1328.60226

Rights

#### Citation

Hilário, Marcelo; den Hollander, Frank; Sidoravicius, Vladas; Soares dos Santos, Renato; Teixeira, Augusto. Random walk on random walks. Electron. J. Probab. 20 (2015), paper no. 95, 35 pp. doi:10.1214/EJP.v20-4437. https://projecteuclid.org/euclid.ejp/1465067201

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