%0 Journal Article
%T A crossover for the bad configurations of random walk in random scenery
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%+ European Institute for Statistics, Probability, Stochastic Operations Research and its Applications (EURANDOM)
%+ Mathematics department
%+ Department of Mathematical Sciences
%A Blachère, Sébastien
%A den Hollander, Frank
%A Steif, Jeffrey E.
%Z Supported in part by EURANDOM in Eindhoven, in part by DFG and NWO through the Dutch-German Bilateral Research Group on “Mathematics of Random Spatial Models from Physics and Biology” (2004–2009) and in part by the Swedish Research Council and by the Göran Gustafsson Foundation for Research in the Natural Sciences and Medicine.
%Z This paper is dedicated to the memory of Oded Schramm.
%< avec comité de lecture
%@ 1050-5164
%J The Annals of Applied Probability
%I Institute of Mathematical Statistics (IMS)
%V 39
%N 5
%P 2018-2041
%8 2011-09
%D 2011
%Z 1103.1780
%R 10.1214/11-AOP664
%K random walk in random scenery
%K conditional probability distribution
%K bad and good configurations
%K large deviations
%Z 60K37; 60G50
%Z Mathematics [math]/Probability [math.PR]Journal articles
%X 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 is an element of [1/2, 1] and epsilon is an element of [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 is an element of (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 is an element of [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 is an element of [1/2, 4/5), which will be pursued in future work.
%G English
%L hal-01311297
%U https://hal.science/hal-01311297
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ I2M