Counting and Sampling Fall 2017 Lecture 5 : Coupling & Strong Stationary Time

We claim that both of these walks have the same mixing time. This fact is in fact true for any Markov chain on groups. In this case note that we are considering a Markov chain on the symmetric group Sn. In general consider any group G and suppose we have a set of generators {g1, . . . , gk}. In each time step we choose a generator from this set according to a probability distribution μ and we apply it to the current state. The inverse chain is defined as follows: Consider the set of generators {g−1 1 , . . . , g −1 k }. At any state x, choose a generator g−1 i from the same distribution μ and apply it to x. Observe that both of these walks are doubly stochastic, so have a uniform stationary distribution.