Mixing time bounds for overlapping cycles shuffles

Consider a deck of n cards. Let p1, p2, . . . , pn be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the pi’s and move the card in that position to the top. This setup was introduced in [5], where a few special cases were studied. In particular the case pn−k = pn = 1/2, k = Θ(n), turned out to be challenging and only a few lower bounds were produced. These were improved in [1] where it was shown that the relaxation time for the motion of a single card is Θ(n2) when k/n approaches a rational number. In this paper we give the first upper bounds. We focus on the case m = bn/2c. It is shown that for the additive symmetrization as well as the lazy version of the shuffle, the mixing time is O(n3 log n). We then consider two other modifications of the shuffle. The first one is the case pn−k = pn−k+1 = 1/4 and pn = 1/2. Using the entropy technique developed by Morris [7], we show that mixing time is O(n5/2 log n) and that for the symmetrized shuffle, O(n2 log n) steps suffice. The second modification is a variant of the first, where the moves are made in pairs so that if the first move involves position n, then the second move must be taken from positions m or m + 1 and vice versa. Interestingly, this shuffle is much slower; the mixing time is at least of order n3 log n and at most of order n3 log n. It is also observed that results of [1] can be modified to improve lower bounds for some k = o(n). Chalmers University of Technology and Göteborg University †http://www.math.chalmers.se/∼jonasson Research supported by the Swedish Research Council