Effective Polynomial Ballisticity Condition for Random Walk in Random Environment

The conditions $(T)_\gamma,$ $\gamma \in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. Among others, these conditions entail a ballistic behaviour as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_\gamma$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type. In addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than $(T)_\gamma,$ another advantage of the condition $(\mathcal{P})$ is that it is effective in the sense that it can be checked on finite boxes. In particular, this extends the conjectured equivalence of the conditions $(T)_\gamma,$ $\gamma \in (0,1),$ to all relevant dimensions.Comment: 21 pages, 2 figures; followed referee's and readers' comments, corrected minor errors; to appear in Comm. Pure Appl. Mat

Sharp ellipticity conditions for ballistic behavior of random walks in random environment

We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ram\'{\i}rez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ram\'{\i}rez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension $d=2$ under which the corresponding random walk is ballistic.Comment: Published at http://dx.doi.org/10.3150/14-BEJ683 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Transition from the annealed to the quenched asymptotics for a random walk on random obstacles

In this work we study a natural transition mechanism describing the passage from a quenched (almost sure) regime to an annealed (in average) one, for a symmetric simple random walk on random obstacles on sites having an identical and independent law. The transition mechanism we study was first proposed in the context of sums of identical independent random exponents by Ben Arous, Bogachev and Molchanov in [Probab. Theory Related Fields 132 (2005) 579--612]. Let $p(x,t)$ be the survival probability at time $t$ of the random walk, starting from site $x$, and let $L(t)$ be some increasing function of time. We show that the empirical average of $p(x,t)$ over a box of side $L(t)$ has different asymptotic behaviors depending on $L(t)$. T here are constants $0<\gamma_1<\gamma_2$ such that if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_1$, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_2$, also a central limit theorem is satisfied. If ${L(t)\ll t}$, the averaged survival probability decreases like the quenched survival probability. If $t\ll L(t)$ and $\log L(t)\ll t^{d/(d+2)}$ we obtain an intermediate regime. Furthermore, when the dimension $d=1$ it is possible to describe the fluctuations of the averaged survival probability when $L(t)=e^{\gamma t^{d/(d+2)}}$ with $\gamma<\gamma_2$: it is shown that they are infinitely divisible laws with a L\'{e}vy spectral function which explodes when $x\to0$ as stable laws of characteristic exponent $\alpha<2$. These results show that the quenched and annealed survival probabilities correspond to a low- and high-temperature behavior of a mean-field type phase transition mechanism.Comment: Published at http://dx.doi.org/10.1214/009117905000000404 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org