Information loss in riffle shuffling

We study the asymptotic behaviour of the relative entropy (to stationarity) for a commonly used model for riffle shuffling a deck of n cards m times. Our results establish and were motivated by a prediction in a recent numerical study of Trefethen and Trefethen. Loosely speaking, the relative entropy decays approximately linearly (in m) for in <log(2)n, and approximately exponentially for m > log(2) n. The deck becomes random in this information-theoretic sense after m = 3/2 log(2) n shuffles.