  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.
