Biased random walk in positive random conductances on Zd\mathbb{Z}^{d}

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

Phase transition for the speed of the biased random walk on the supercritical percolation cluster

We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, 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 sub-ballistic 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.Comment: 78 pages with 16 figures. Changes made at the suggestion of a referee, including more detail in Section 8. Comm. Pure Appl. Math., accepte

Biased random walks on random graphs

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.Comment: Survey based one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. 64 pages, 16 figure

Biased random walk on critical Galton-Watson trees conditioned to survive

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs

Quenched invariance principle for biased random walks in random conductances in the sub-ballistic regime

We consider a biased random walk in positive random conductances on $\mathbb{Z}^d$ for $d\geq 5$. In the sub-ballistic regime, we prove the quenched convergence of the properly rescaled random walk towards a Fractional Kinetics.Comment: 59 pages, 8 figures. Modifications in Theorem 1.2, Section 3-4-