Random walk on random walks: higher dimensions

TL;DR

This paper extends the analysis of a random walk in dynamic random environments to higher dimensions and more general transition kernels, establishing laws of large numbers, central limit theorems, and large deviations.

Contribution

It introduces new renormalization and renewal techniques to handle non-monotonicity in higher dimensions and general kernels, broadening previous results.

Findings

Established strong law of large numbers for the walker's position.

Proved a functional central limit theorem in high-density regimes.

Derived large deviation estimates for the walker's position.

Abstract

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].