Stable fluctuations for ballistic random walks in random environment on Z
Abstract
We consider transient random walks in random environment on $\Z$ in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following [7], where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. Furthermore, our proof enables us to give a description of the quenched distribution of hitting times. The case of Dirichlet environment turns out to be remarkably explicit.
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