Cutoff for random to random card shuffle

Abstract

In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4}n\log n-\frac{1}{4}n\log \log n$ with window of order $n$, answering a conjecture of Diaconis.