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, $I_a\le I_q$. 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 $d\ge 4$.

21 pages. In this revised version, we corrected our computation of the variance of $D(B_1)$ for $d=2+1$ (page 11 of the old version, after (2.31)). We also added details explaining precisely how the space-only case is handled, by mapping the appropriate objects to the space-time setup (see pages 14--17 in the new version). Accepted for publication in Communications in Mathematical Physics.

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