Limiting behavior for the distance of a random walk

In this paper we study some aspects of the behavior of random walks on large but finite graphs before they have reached their equilibrium distribution. This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time n/2. Here, we study the examples of random 3-regular graphs, random adjacent transpositions, and riffle shuffles. In the case of a random 3-regular graph, there is a phase transition where the speed changes from 1/3 to 0 at time 3 log(2) n. A similar result is proved for riffle shuffles, where the speed changes from 1 to 0 at time log(2) n. Both these changes occur when a distance equal to the average diameter of the graph is reached. However in the case of random adjacent transposition, the behavior is more complex. We find that there is no phase transition, even though the distance has different scalings in three different regimes.