Improved mixing time bounds for the Thorp shuffle

TL;DR

This paper proves that the Thorp shuffle, a card shuffling process, mixes in logarithmic cubic time for decks of size a power of two, improving previous bounds from four to three logarithmic factors.

Contribution

The paper establishes a tighter upper bound of O(log^3 n) on the mixing time of the Thorp shuffle for decks of size a power of two, enhancing prior results.

Findings

Mixing time is O(log^3 n) for n a power of two.

Previous bound was O(log^4 n).

Results improve understanding of the Thorp shuffle's efficiency.

Abstract

E. Thorp introduced the following card shuffling model. 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 show that if $n$ is a power of 2 then the mixing time of the Thorp shuffle is $O(\log^3 n)$. Previously, the best known bound was $O(\log^4 n)$.