Random Walk on Random Walks

TL;DR

This paper investigates the behavior of a random walk in a dynamic environment of particles, establishing laws of large numbers, central limit theorem, and large deviations under certain conditions.

Contribution

It introduces a novel analysis of a random walk in a dynamic random environment, proving fundamental probabilistic limit theorems with a new renewal and renormalization approach.

Findings

Strong law of large numbers for the walk's position

Functional central limit theorem established

Large deviation bounds proved for large densities

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.