Random walks on random walks: non-perturbative results in high dimensions

TL;DR

This paper establishes classical limit theorems, including a law of large numbers and a central limit theorem, for a random walk in a dynamic environment generated by independent particles, specifically in high dimensions.

Contribution

It provides non-perturbative, dimension-dependent results for the model, extending previous work and contrasting low-dimensional anomalous behaviors.

Findings

Proves strong law of large numbers for d ≥ 5

Establishes large deviation estimates for d ≥ 5

Derives a functional central limit theorem for d ≥ 9

Abstract

Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$ whose evolution at time $t$ depends on the number of such particles at its location. We derive classical limit theorems for this instrumental model of a random walk in a dynamic random environment, applicable in sufficiently high dimensions. More precisely, for $d \geq 5$, we prove a strong law of large numbers and large deviation estimates. Further, for $d\geq 9$, we obtain a functional central limit theorem under the annealed law. These results are non-perturbative in the sense that they hold for any positive density of the Poissonian field. Under the aforementioned assumptions on the dimension they therefore improve on previous work on the model. Moreover, they stand in contrast to the anomalous behaviour predicted in low dimensions.