Information Loss in Riffle Shuffling

Source abstract

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 m < 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.