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

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