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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 Folner property almost surely

Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times