Mixing Time of the Rudvalis Shuffle

Abstract

We extend a technique for lower-bounding the mixing time of card-shuffling Markov chains, and use it to bound the mixing time of the Rudvalis Markov chain, as well as two variants considered by Diaconis and Saloff-Coste. We show that in each case $\Theta(n^3 \log n)$ shuffles are required for the permutation to randomize, which matches (up to constants) previously known upper bounds. In contrast, for the two variants, the mixing time of an individual card is only $\Theta(n^2)$ shuffles.