Improved mixing time bounds for the Thorp shuffle

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 $O(\log^4 n)$.