Scaling of sub-ballistic 1D Random Walks among biased Random Conductances

Abstract

We consider two models of one-dimensional random walks among biased i.i.d. random conductances: the first is the classical exponential tilt of the conductances, while the second comes from the effect of adding an external field to a random walk on a point process (the bias depending on the distance between points). We study the case when the walk is transient to the right but sub-ballistic, and identify the correct scaling of the random walk: we find α P r0, 1s such that log Xn{ log n Ñ α. Interestingly, α does not depend on the intensity of the bias in the first case, but it does in the second case.

Regularity of biased 1D random walks in random environment

“…The behavior of one dimensional RWRE's that are transient but with zerospeed has been studied in [8,9,20] for i.i.d. jump probabilities and in [4] for the RCM. The continuity of the diffusion matrix at λ = 0 has been derived in [29] for diffusions in random environment.…”

“…The random conductance model (RCM), which is of particular interest in what follows, appears in Section 2. 4.…”