Random walk in cooling random environment: recurrence versus transience and mixed fluctuations.

This is the third in a series of papers in which we consider a one-dimensional model of Random Walk in Cooling Random Environment (RWCRE), obtained by starting from Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In two earlier papers, in the regime where the increments of the refreshing times diverge, we derived a strong law of large numbers and a large deviation principle under the quenched measure. Both the speed and the rate function turned out to be the same as for RWRE. We also derived a centered central limit theorem under the annealed measure for the case where RWRE is recurrent and the refreshing times have either polynomial or exponential growth. Both the scale and the limit law turned out to be different from those of RWRE. In the present paper we address the question of recurrence versus transience. In addition, we explore fluctuations for general refreshing times when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. In particular, sub-diffusive behaviour and convergence to mixtures of different limit laws can occur. We conclude by briefly commenting on possible extensions and open problems.