Abstract
LetG be an infinite connected graph with vertex setV. Ascenery onG is a map ξ :V → 0, 1 (equivalently, an assignment of zeroes and ones to the vertices ofG). LetS n n≥0 be a simple random walk onG, starting at some distinguished vertex v0. Now let ξ and η be twoknown sceneries and assume that we observe one of the two sequences ξ(S n) n≥0 or {η(S n)} n≥0 but we do not know which of the two sequences is observed. Can we decide, with a zero probability of error, which of the two sequences is observed? We show that ifG = Z orG = Z2, then the answer is “yes” for each fixed ξ and “almost all” η. We also give some examples of graphsG for which almost all pairs (ξ, η) are not distinguishable, and discuss some variants of this problem.
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The work of I.B. was supported by the US. Army Research Office through the Mathematical Sciences Institute of Cornell University. The work of H.K. was supported by the NSF through a grant to Cornell University.
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Benjamini, I., Kesten, H. Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69, 97–135 (1996). https://doi.org/10.1007/BF02787104
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DOI: https://doi.org/10.1007/BF02787104