Scaling of Sub-Ballistic 1D Random Walks Among Biased Random Conductances

Q. Berger, M. Salvi

2019, v.25, Issue 1, 171-187

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 $\ga \in[0,1]$ such that $\log X_n /\log n \to \ga$. Interestingly, $\ga$ does not depend on the intensity of the bias in the first case, but it does in the second case. %Moreover, with additional information on the distribution of the conductances, we are able to identify more sharply the correct scaling, and the limiting distribution for the rescaled $X_n$.

Keywords: random walk, random environment, limit theorems, conductance model, Mott random walk

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