Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three

Abstract

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, $I_a\leq I_q$. In the space-time case, when $d\geq3+1$, $I_q$ 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 $d\geq4$.