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

TL;DR

This paper investigates the scaling behavior of one-dimensional biased random walks in random conductances, revealing how the walk's growth rate depends on the bias and conductance models, especially in sub-ballistic regimes.

Contribution

It identifies the precise scaling exponent for sub-ballistic walks among biased conductances and shows how this exponent varies with the bias in different models.

Findings

The scaling exponent  is determined for sub-ballistic walks.

In the first model,  is independent of bias intensity.

In the second model,  depends on the bias.

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 $\alpha \in[0,1]$ such that $\log X_n / \log n \to \alpha$. Interestingly, $\alpha$ does not depend on the intensity of the bias in the first case, but it does in the second case.