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 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP664 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit theorems for additive functionals of random walks in random scenery

“…This dynamical system preserves the infinite measure µ := P ⊗Z X 1 ⊗ P ⊗Z ξ 0 ⊗ λ Z , where λ Z is the counting measure on Z. Actually, thanks to (5) and to the recurrence ergodicity of ( Ω, T , µ), we prove the following stronger version of the convergence in distribution of Theorem 3.…”

“…The proof of the moments convergence in Theorem 3 is a straigthforward adaptation of [14] and is given in Appendix B. Due to Theorem 1 and to the above argument that lead to (5), the convergence in distribution in Theorem 3 is a consequence of the moments convergence. Another strategy to prove the convergence in distribution in Theorem 3 consists in seing this result as a direct consequence of (5) combined with Proposition 13 stating the ergodicity of the dynamical system ( Ω, T , µ) corresponding to…”

“…In Section 2, we prove Theorem 1 (bounds on moments of the local time of the Kesten Spitzer process) and Theorem 2 (estimate on the distance in L 2 (R) between the local time of a Brownian motion and a k-dimensional vector space). In Section 3, we establish the recurrence ergodicity of the infinite measure preserving dynamical system ( Ω, T , µ) and obtain the convergence in distribution of Theorem 3 (Law of Large Numbers) as a byproduct of this recurrence ergodicity combined with (5). Section 3 is completed by Appendix B which contains the proof of the moments convergence of Theorem 3.…”

Generalized Divide and Color Models

“…We describe how we came up with the example for the n = 6 case. The negations of conditions (i) and (ii) of Lemma 2.2 for S = [6] give a set of linear equations that must hold in order for two RERs to have the same color process. With the help of Mathematica, the nullspace of the coefficient matrix of the linear system was calculated.…”

“…Remark 7.16. In [6], a phase transition in σ is shown for random walk in random scenery, concerning Gibbsianness of the process. Is it possible that this could be related to a phase transition concerning the stochastic domination behavior?…”