Relative Complexity of Random Walks in Random Scenery in the absence of a weak invariance principle for the local times

TL;DR

This paper investigates the complexity of random walks in random scenery driven by various types of two-dimensional walks, establishing properties like the F"olner property of the walk's range.

Contribution

It provides new insights into the relative complexity of different random walks in random scenery, especially without relying on a weak invariance principle for local times.

Findings

Range of the random walk satisfies the F"olner property almost surely

Analyzes complexity for walks in the domain of attraction of the Cauchy distribution

Addresses the question posed by Aaronson about relative complexity

Abstract

We answer the question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the F\"olner property almost surely.