Abstract
We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z} ^{d}$, for $d\geq 2$. This extends the results from 2003 established by Sznitman in [12] and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow’s condition, through a new sharp version of Kalikow’s criteria. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coarse-graining methods together with a variation of the Azuma-Hoeffding concentration inequality in order to verify the fulfillment of a ballisticity condition by Berger, Drewitz and Ramírez.
Highlights
Introduction and main results1.1 IntroductionThe random walk in a random environment is one of the fundamental models describing the movement of a particle in disordered media
For walks on Zd with d ≥ 2, few results exist giving explicit formulas for basic associated quantities such as the velocity, asymptotic direction or variance, or conditions characterizing specific long-term behavior such as transience/recurrence, directional transience and ballistic movement. It is still a widely open problem to explicitly characterize the small disorder necessary to produce ballistic behavior whenever added to the jump probabilities of the simple symmetric
In this article we focus on this particular question and generalize previously known conditions by exploring the use of refined concentration inequalities which are variations of the well-known Azuma-Hoeffding inequality
Summary
The random walk in a random environment is one of the fundamental models describing the movement of a particle in disordered media (see [13, 14] for a comprehensive overview of the model). For walks on Zd with d ≥ 2, few results exist giving explicit formulas for basic associated quantities such as the velocity, asymptotic direction or variance, or conditions characterizing specific long-term behavior such as transience/recurrence, directional transience and ballistic movement. It is still a widely open problem to explicitly characterize the (law of the) small disorder necessary to produce ballistic behavior whenever added to the jump probabilities of the simple symmetric. Before we state our results more precisely and give further details/discussion, let us formally introduce the model and set up the framework to be used throughout the article
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