Electronic Journal of Probability

Random walk on random walks

Marcelo Hilário, Frank den Hollander, Vladas Sidoravicius, Renato Soares dos Santos, and Augusto Teixeira

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82C22: Interacting particle systems [See also 60K35] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Random walk dynamic random environment strong law of large numbers functional central limit theorem large deviation bound Poisson point process coupling renormalisation regeneration times

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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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