Correction to "The stepping stone model in a random environment" -- limit theorems for the occupation time and intersection time of reversible random walks in random environments

Abstract

This note corrects a mistake in the author's above-mentionned published work regarding the intersection local time of two independent random walks in a random environment. The random walks each behave as nearest neighbour random walks in random conductances in the same random environment. The proof of the main result in the original work relied on the derivation of the scaling limit of an additive functionnal of the environment at the locations where the two independent random walks meet, which was shown to be given by a constant times the intersection local time of two Brownian motions. This note shows how the value of the constant denoted by ${\hat{I}}^3$ in this work should be updated to correct the statement. The derivation of this constant uses new results on the scaling limit of the occupation time of reversible random walks in a random environment.