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 ≤ 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.
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Acknowledgments
This research was supported partially by a grant from the Israeli Science Foundation, and by the Alhadeff Fund at theWeizmann Institute.We thank Francis Comets for providing us with an update on polymer models and bringing the work of Lacoin [10] to our attention.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Yilmaz, A., Zeitouni, O. Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three. Commun. Math. Phys. 300, 243–271 (2010). https://doi.org/10.1007/s00220-010-1119-3
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DOI: https://doi.org/10.1007/s00220-010-1119-3