Limit theorems for random walks on a strip in subdiffusive regimes

Abstract

We study the asymptotic behaviour of occupation times of a transient random walk (RW) in a quenched random environment (RE) on a strip in a subdiffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is exactly the same. As a particular case, we solve a long standing problem of describing the asymptotic behaviour of a RW with bounded jumps on a one-dimensional lattice. Our technique results from the development of ideas from our previous work (Dolgopyat and Goldsheid 2012 Commun. Math. Phys. 315 241-77) on the simple RWs in RE and those used in Bolthausen and Goldsheid (2000 Commun. Math. Phys. 214 429-47; 2008 Commun. Math. Phys. 278 253-88) and Goldsheid (2008 Probab. Theory Relat. Fields 141 471-511) for the study of random walks on a strip. © 2013 IOP Publishing Ltd & London Mathematical Society.