No abstract

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ℤd. That is, for each d ≥ 2, and for any supercritical parameter p > pc, we prove the existence of a critical strength for the bias such that below this value the speed is positive, and above the value it is zero. We identify the value of the critical bias explicitly, and in the subballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one‐dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known. © 2013 Wiley Periodicals, Inc.

The speed v.ˇ/ of aˇ-biased random walk on a Galton-Watson tree without leaves is increasing forˇ 1160.

We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting.Comment: Published in at http://dx.doi.org/10.1214/13-AOP835 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in (0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.Comment: 43 pages, 4 figures; to appear in Probability Theory and Related Field