E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract Suppose that the vertices of the lattice 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

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... alks 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. Detection of RW trails and let Q X be the product measure Q the labels off [X ] are sampled from µ and the labels on [X ] are sampled from ν.) Finally, define the Borel measure Q on Ω V (G) by We say that the distributions P and Q are indistinguishable if P and Q are absolutely continuous with respect to each other; otherwise, we say that P and Q are distinguishable. In general, examples exist of measures P, Q, constructed as above, that are distinguishable but not singular. However, throughout most of this paper we choose to focus on graphs G and path distributions Ψ where this intermediate situation does not occur. Indeed, this can be established when Ψ is the law of an automorphisminvariant Markov chain on a transitive graph. Proposition 1.1. Let G be a transitive graph and let M be a transition kernel on V (G) which is invariant under a transitive subgroup H of automorphisms of G. (That is, M (h(x), h(y)) = M (x, y) for x, y ∈ V (G) and h ∈ H.) Let Ψ be the law of the Markov chain with transition law M and initial state v; we assume that this chain is transient. Then the measures P and Q are either singular, or mutually absolutely continuous. Proof of Part 1 of Theorem 1.2. Since µ = ν, there exists some ρ ∈ Ω with ν(ρ) > µ(ρ). Let k(n) = (log n) α with α = 1/(d − 1). The singularity of P and Q follows from the following claim.