Distinguishing sceneries by observing the scenery along a random walk path

Let g be an infinite connected graph with vertex set nu. A scenery on g is a map xi : nu --> {0, 1} (equivalently, an assignment of zeroes and ones to the vertices of g). Let {S-n}(n greater than or equal to 0) be a simple random walk on g, starting at some distinguished vertex nu(0). Now let xi and eta be two known sceneries and assume that we observe one of the two sequences {xi(S-n)}(n greater than or equal to 0) or {eta(S-n)}(n greater than or equal to 0), but we do not know which of the two sequences is observed. Can we decide, with a zero probability of error, which of the two sequences is observed? We show that if g = Z or g = Z(2), then the answer is ''yes'' for each fixed xi and ''almost all'' eta. We also give some examples of graphs g for which almost all pairs (xi, eta) are not distinguishable, and discuss some variants of this problem.