## Electronic Journal of Probability

### Detecting the trail of a random walker in a random scenery

#### 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 />

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1465064312

Digital Object Identifier
doi:10.1214/EJP.v18-2367

Mathematical Reviews number (MathSciNet)
MR3119085

Zentralblatt MATH identifier
1290.60050

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

Rights

#### Citation

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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