Random walk on random walks Article Swipe
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· 2015
· Open Access
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· DOI: https://doi.org/10.1214/ejp.v20-4437
· OA: W2139153359
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.
