Suppose that the integers are assigned i.i.d. random variables f! x g (taking values in the unit interval), which serve as an environment. This environment deenes a random walk fX k g (called a RWRE) which, when at x, moves one step to the right with probability ! x , and one step to the left with probability 1 ? ! x. Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) ...

In this paper the author continues his investigation into the scaling limit of a partial difference equation on the d-dimensional integer lattice Zd, corresponding to a translation invariant random walk perturbed by a random vector field. In a previous paper he obtained a formula for the effective diffusion constant. It is shown here that for the nearest neighbor walk in dimension d ≥ 3 this ef...

We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree Γ are i.i.d. random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on Γ...

We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on Z = {0, 1, 2, . . .}, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different...

We consider random walk in elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the bal-listicity condition (T ′). We show that for every ǫ > 0 and n large enough, the annealed probability of linear slowdown is bounded above by exp`− (log n) d−ǫ´. This is almost matching the known lower bound of exp`− C(log n) d ´ , and significantly improves previously known u...

In this paper, the author investigates the scaling limit of a partial difference equation on the d dimensional integer lattice Zd, corresponding to a translation invariant random walk perturbed by a random vector field. In the case when the translation invariant walk scales to a Cauchy process he proves convergence to an effective equation on Rd. The effective equation corresponds to a Cauchy p...