The Annals of Probability

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

Sébastien Blachère, Frank den Hollander, and Jeffrey E. Steif

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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.

Article information

Ann. Probab., Volume 39, Number 5 (2011), 2018-2041.

First available in Project Euclid: 18 October 2011

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Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Random walk in random scenery conditional probability distribution bad and good configurations large deviations


Blachère, Sébastien; den Hollander, Frank; Steif, Jeffrey E. A crossover for the bad configurations of random walk in random scenery. Ann. Probab. 39 (2011), no. 5, 2018--2041. doi:10.1214/11-AOP664.

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  • [1] den Hollander, F. (2000). Large Deviations. Fields Institute Monographs 14. Amer. Math. Soc., Providence, RI.
  • [2] den Hollander, F. and Steif, J. E. (2006). Random walk in random scenery: A survey of some recent results. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 48 53–65. IMS, Beachwood, OH.
  • [3] den Hollander, F., Steif, J. E. and van der Wal, P. (2005). Bad configurations for random walk in random scenery and related subshifts. Stochastic Process. Appl. 115 1209–1232.
  • [4] Lőrinczi, J., Maes, C. and Vande Velde, K. (1998). Transformations of Gibbs measures. Probab. Theory Related Fields 112 121–147.
  • [5] Maes, C., Redig, F., Van Moffaert, A. and Leuven, K. U. (1999). Almost Gibbsian versus weakly Gibbsian measures. Stochastic Process. Appl. 79 1–15.
  • [6] Spitzer, F. (1976). Principles of Random Walks, 2nd ed. Graduate Texts in Mathematics 34. Springer, New York.
  • [7] van Enter, A. C. D., Le Ny, A. and Redig, F. (eds.) (2004). Gibbs versus non-Gibbs in Statistical Mechanics and Related Fields (Proceedings of a Workshop at EURANDOM, Eindhoven, The Netherlands, December 2003). Markov Process. Related Fields 10 377–564.