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In this paper, we consider a random walk and a random color scenery on Z. Theincrements 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 ofcolors seen by the walk in the course of time. Bad configurations for thisrandom process are the discontinuity points of the conditional probabilitydistribution for the color seen at time zero given the colors seen at all latertimes. We focus on the case where the random walk has increments 0, +1 or -1with probability epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, withp in [1/2,1] and epsilon in [0,1), and where the scenery assigns the colorblack 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=0and p in (1/2,4/5) all configurations are bad, while for (p,epsilon) in an openneighborhood of (1,0) all configurations are good. In addition, we show thatfor epsilon=0 and p=1/2 both bad and good configurations exist. We conjecturethat 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 casewhere epsilon>0 and p in [1/2,4/5), which will be pursued in future work.

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