Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment

Details Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment

2010-05-03

arxiv.org/abs/1005.0376v2

Annals of Probability 2012, Vol. 40, No. 2, 459-534

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP637 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP637

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $\gamma\in(0,1)$ the ballisticity condition $(T)_{\gamma}$ and the condition $(T')$ defined as the fulfillment of $(T)_{\gamma}$ for each $\gamma\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $\gamma\in (0.5,1)$, $(T)_{\gamma}$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $\gamma\in (0,1)$, condition $(T)_{\gamma}$ implies a ballistic law of large numbers. On the other hand, Drewitz and Ram\'{{\i}}rez have shown that in dimensions $d\ge2$ there is a constant $\gamma_d\in(0.366,0.388)$ such that for each $\gamma\in(\gamma_d,1)$, condition $(T)_{\gamma}$ is equivalent to $(T')$. Here, for dimensions larger than three, we extend the previous range of equivalence to all $\gamma\in(0,1)$. For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.

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