Abstract
Suppose that the vertices of the lattice $\mathbb{Z}^d$ are endowed with a random scenery, obtained by tossing a fair coin at each vertex. A random walker, starting from the origin, replaces the coins along its path by i.i.d. biased coins. For which walks and dimensions can the resulting scenery be distinguished from the original scenery? We find the answer for simple random walk, where it does not depend on dimension, and for walks with a nonzero mean, where a transition occurs between dimensions three and four. We also answer this question for other types of graphs and walks, and raise several new questions.<br />
Highlights
Let μ and ν be different probability measures on the same finite sample space Ω, such that μ(ω) > 0 and ν(ω) > 0 for all ω ∈ Ω
I.i.d. labels with law μ are attached to the vertices of G. (Call the law of this random scenery P )
An infinite random path X with distribution Ψ is chosen, and the labels along X are replaced by independent labels with law ν; this yields a new random scenery with distribution Q
Summary
Let μ and ν be different probability measures on the same finite sample space Ω, such that μ(ω) > 0 and ν(ω) > 0 for all ω ∈ Ω. The answer depends on the choice of G, Ψ, μ and ν; as we shall see, it is sometimes quite surprising To state this problem formally, let P be the product measure μV (G), which is the initial distribution of the scenery. Throughout most of this paper we choose to focus on graphs G and path distributions Ψ where this intermediate situation does not occur. This can be established when Ψ is the law of an automorphisminvariant Markov chain on a transitive graph. Let Ψ be the law of simple random walk on Zd. for all dimensions d and all μ = ν, the distributions P and Q are singular. Part (2) of Theorem 1.2 is closely related to a result of Bolthausen and Sznitman [5] on random walks in random environment
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