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Detecting the trail of a random Walker in a random scenery. (English) Zbl 1290.60050
Authors’ 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.
##### MSC:
 60G50 Sums of independent random variables; random walks 60K37 Processes in random environments
##### Keywords:
random walk; branching number
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