Random walk on random walks: low densities

TL;DR

This paper studies a random walk in a dynamic environment formed by independent random walks, revealing how low particle density and local drift influence its ballistic behavior, with differences depending on laziness of particles.

Contribution

It introduces new results on ballisticity, laws of large numbers, and limit theorems for random walks in dynamic environments with varying particle densities and drift conditions.

Findings

Ballisticity depends on particle laziness and density.

Strong law of large numbers established for the walk.

Functional CLT and large deviation bounds derived.

Abstract

We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.