Detecting the trail of a random walker in a random scenery

TL;DR

This paper investigates whether a random walker's path can be distinguished from the original random scenery in a lattice, revealing dimension-dependent detectability and extending results to various graphs and walks.

Contribution

It provides a comprehensive analysis of scenery detectability for simple and biased random walks across different dimensions and graph types.

Findings

Detection does not depend on dimension for simple random walk.

A phase transition occurs between dimensions three and four for walks with nonzero mean.

Results extend to other graph structures and walk types.

Abstract

Suppose that the vertices of the Euclidean 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 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.