Electronic Journal of Probability

Detecting the trail of a random walker in a random scenery

Noam Berger and Yuval Peres

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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 />

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 87, 18 pp.

Accepted: 3 October 2013
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K37: Processes in random environments

Random walk Random scenery Relative entropy Branching number

This work is licensed under a Creative Commons Attribution 3.0 License.


Berger, Noam; Peres, Yuval. Detecting the trail of a random walker in a random scenery. Electron. J. Probab. 18 (2013), paper no. 87, 18 pp. doi:10.1214/EJP.v18-2367. https://projecteuclid.org/euclid.ejp/1465064312

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