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 ρ ∈ (0; ¥). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p ● when it is on a vacant site and probability p ● when it is on an occupied site. Assuming that p ○ ∈ (0, 1) and p ● ≠ 1/2 , 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 ρ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.