Rapid mixing of dealer shuffles and clumpy shuffles

Clumpy Riffle Shuffles

Sticky or clumpy riffle shuffles appear quite naturally in applications; however, the problem of precisely describing the convergence rate of such modified riffle shuffles has remained open for a long time. In this paper, we develop an alternative family of clumpy shuffles, and analyze their convergence rate. We then provide empirical evidence that our alternative shuffles converge at a similar...

Mixing Time of the Rudvalis Shuffle

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 Θ(n log n) shuffles are required for the permutation to randomize, which matches (up to constants) previously known upper bounds. In contrast, for the...

The Use of Single Position Observations in Evaluating Mixing T imes of Card Shuffles

Riffle shuffles of decks with repeated cards

By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11 — with the former number applying when the metric used to define mixing is the total variation distance and the later when it is the separation distance. This and other related results assum...

Analysis of Top to Bottom - K Shuffles 3

A deck of n cards is shuffled by repeatedly moving the top card to one of the bottom kn positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as kn ranges from a constant to n. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom kn positions or a random car...