A crossover for the bad configurations of random walk in random scenery

Abstract

In this paper, we consider a random walk and a random color scenery on ℤ. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times.

We focus on the case where the random walk has increments 0, +1 or −1 with probability ε, (1 − ε)p and (1 − ε)(1 − p), respectively, with p ∈ [½, 1] and ε ∈ [0, 1), and where the scenery assigns the color black or white to the sites of ℤ with probability ½ each. We show that, remarkably, the set of bad configurations exhibits a crossover: for ε = 0 and p ∈ (½, ⅘) all configurations are bad, while for (p, ε) in an open neighborhood of (1, 0) all configurations are good. In addition, we show that for ε = 0 and p = ½ both bad and good configurations exist. We conjecture that for all ε ∈ [0, 1) the crossover value is unique and equals ⅘. Finally, we suggest an approach to handle the seemingly more difficult case where ε > 0 and p ∈ [½, ⅘), which will be pursued in future work.