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, $I_a\le I_q$. In the space-time case, when $d\ge 3+1$, $I_q$ and $I_a$ are known to be equal on an open set containing the typical velocity ${\xi }_o$. When $d=1+1$, we prove that $I_q$ and $I_a$ are equal only at ${\xi }_o$. Similarly, when d=2+1, we show that $I_a<I_q$ on a punctured neighborhood of ${\xi }_o$. 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 ${\xi }_o$ whenever the walk is in that class. This is very different from the known results for non-nestling walks on ${\mathbb{Z}}^d$ with $d\ge 4$.