Central Limit Theorem for Recurrent Random Walks on a Strip with Bounded Potential

We prove that a recurrent random walk (RW) in i.i.d. random environment (RE) on a strip which does not obey the Sinai law exhibits the Central Limit asymptotic behaviour. We also show that there exists a collection of proper subvarieties in the space of transition probabilities such that • If RE is stationary and ergodic and the transition probabilities are concentrated on one of subvarieties from our collection then the CLT holds; • If the environment is i.i.d then the above condition is also necessary for the CLT. As an application of our techniques we prove the CLT for the quasiperiodic environments with Diophantine frequencies in the recurrent case and complement this result by proving that in the transient case the CLT holds for all strictly ergodic environments. All these results are valid for one-dimensional RWRE with bounded jumps as a particular case of the strip model. 2000 Mathematics Subject Classification: primary 60K37, 60F05; secondary 60J05, 82C44.