Improved mixing time bounds for the Thorp shuffle and L-reversal chain

Abstract

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models.

E. Thorp introduced the following card shuffling model in 1973: Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log⁴n). Previously, the best known bound was O(log²⁹n) and previous proofs were only valid for n a power of 2.

We also analyze the following model, called the L-reversal chain, introduced by Durrett: There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n³/L³)log n). We obtain a bound that is within a factor O(log²n) of this, the first bound within a poly log factor of the conjecture.