We will see how a problem in genome rearrangement leads to describe a new kind of phase transition for random walks on graphs. This phase transition is related to the well-known Erdos-Renyi double jump phenomenon for random graphs. I will particularly try to describe the effect that the scale of mutations may have on the analysis of the problem, and will outline two possible approaches: one borrowing ideas from hyperbolic geometry and the other based on cheating and using branching random walks.