We answer a 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 Fölner property almost surely.
"Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times." Ann. Probab. 45 (4) 2505 - 2532, July 2017. https://doi.org/10.1214/16-AOP1118