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

TL;DR

This paper investigates the conditions under which the process of colors seen by a random walk in a random scenery exhibits discontinuities, revealing a crossover phenomenon depending on the walk's bias and scenery parameters.

Contribution

It identifies a crossover in the set of bad configurations for the random walk in random scenery, depending on parameters epsilon and p, and conjectures the uniqueness of this crossover point.

Findings

For epsilon=0 and p in (1/2,4/5), all configurations are bad.

Near (p,epsilon)=(1,0), all configurations are good.

At epsilon=0 and p=1/2, both good and bad configurations exist.

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 epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, with p in [1/2,1] and epsilon in [0,1), and where the scenery assigns the color black or white to the sites of Z with probability 1/2 each. We show that, remarkably, the set of bad configurations exhibits a crossover: for epsilon=0 and p in (1/2,4/5) all configurations are bad, while for (p,epsilon) in an open neighborhood of (1,0) all configurations are good. In addition, we show that for epsilon=0 and p=1/2 both bad and good configurations exist. We conjecture that for all epsilon in [0,1) the crossover value is unique and equals 4/5. Finally, we suggest an approach to handle the seemingly more difficult case where epsilon>0 and p in [1/2,4/5), which will be pursued in future work.