Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three

We consider the quenched and the averaged (or annealed) large deviation rate functions I q and I a for space-time and (the usual) space-only RWRE on $${\mathbb{Z}^d}$$ . By Jensen’s inequality, I a ≤ I q . In the space-time case, when d ≥ 3 + 1, I q and I a are known to be equal on an open set containing the typical velocity ξ o . When d = 1 + 1, we prove that I q and I a are equal only at ξ o . Similarly, when d = 2 + 1, we show that I a < I q on a punctured neighborhood of ξ o . In the space-only case, we provide a class of non-nestling walks on $${\mathbb{Z}^d}$$ with d = 2 or 3, and prove that I q and I a are not identically equal on any open set containing ξ o whenever the walk is in that class. This is very different from the known results for non-nestling walks on $${\mathbb{Z}^d}$$ with d ≥ 4