2 Mixing of Riffle Shuffle via Strong Stationary Times

1.2 Relating the inverse and standard shuffles We now need to relate ∆(t) to ∆(t). We will do this by taking a more general view of shuffles. We will think of a shuffle as a random walk on the symmetric group Sn. The walk is defined by a set of generators G = {g1, . . . , gk} and a probability distribution P over G. (At each step of the walk, we pick a random gi according to P and apply gi to the current state.) Since each generator in G has an inverse in Sn, we see that the random walk process is doubly-stochastic. Therefore, the stationary distribution of the walk is uniform over Sn. We define the inverse shuffle by the generator set G ′ = {g−1 1 , . . . , g −1 k } with the probability distribution P ′(g−1 i ) = P (gi). Claim 1. ∆(t) = ∆(t).