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.
Highlights
Introduction and main resultsBackgroundRandom motion in a random medium is a topic of major interest in mathematics, physics andchemistry
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
Since the pioneering work of Harris [14], there has been much interest in studies of random walk in random environment within probability theory, both for static and dynamic random environments, and a number of deep results have been proven for various types of models
Summary
Random motion in a random medium is a topic of major interest in mathematics, physics and (bio-)chemistry. The random walk trajectory is a concatenation of “large independent random pieces”, and this forms the basis on which the limit laws in Theorem 1.2 can be deduced (after appropriate tail estimates) None of these techniques are new in the field, but in the context of slow mixing dynamic random environments they are novel and open up gates to future advances. This lower speed of the random walk plays the role of a ballisticity condition and is crucial, where we introduce a random sequence of regeneration times at which the random walk “refreshes its outlook on the random environment”, and show that these regeneration times have a good tail. Appendices A–E collect a few technical facts that are needed along the way
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