Detecting a single defect in a scenery by observing the scenery along a random walk path

Summary

A scenery on ℤ is a map ξ: ℤ → {0,..., k − 1}; we think of ξ as a coloring of ℤ, which assigns to each point of ℤ one of k colors. For a given scenery ξ, denote by $\hat{\xi }$ a scenery obtained from ξ by changing ξ(0) only. Let {S _n}_n≥0 be a simple symmetric random walk on ℤ, starting at the origin. Assume that we observe one of the two sequences {ξ(S _n)}_n≥0 or $\hat{\xi }$(S _n)}_n≥0, without being told which of the two sequences is observed. If ξ is known, can we decide (with zero probability of error) on the basis of these observations which of the two sequences was observed ? We prove that this can be done for ‘almost all’ ξ, when k≥5.