Let G be a transient graph, and flip a fair coin at each vertex.
This gives a distribution P. Now start a random walk from a vertex v, and
retoss the coin at each visited vertex, this time with probability 0.75
for heads and probability 0.25 for tails. The eventual configuration of
the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t.
each other? are they singular? (i.e. can you tell whether a random walker
had tampered with the coins or not?) In the talk I'll answer to this
question for various graphs and various types of random walk. Based on
joint work with Y. Peres.