Hitting time mixing for the random transposition walk

TL;DR

This paper proves that the time until all cards are touched in a random transposition shuffle closely matches the cutoff time for mixing, confirming a conjecture and establishing a hitting time perspective.

Contribution

It establishes the hitting time version of the cutoff phenomenon for the random transposition shuffle, confirming a conjecture by Berestycki.

Findings

Total variation distance at hitting time $ au$ is $o_n(1)$.

At time $ au-1$, the total variation distance is at least $(1+o_n(1))e^{-1}$.

Hitting all cards is asymptotically optimal for mixing time.

Abstract

Consider shuffling a deck of $n$ cards, labeled $1$ through $n$, as follows: at each time step, pick one card uniformly with your right hand and another card, independently and uniformly with your left hand; then swap the cards. How long does it take until the deck is close to random? Diaconis and Shahshahani showed that this process undergoes cutoff in total variation distance at time $t = \lfloor n\log{n}/2 \rfloor$. Confirming a conjecture of N.~Berestycki, we prove the definitive ``hitting time'' version of this result: let $\tau$ denote the first time at which all cards have been touched. The total variation distance between the stopped distribution at $\tau$ and the uniform distribution on permutations is $o_n(1)$; this is best possible, since at time $\tau-1$, the total variation distance is at least $(1+o_n(1))e^{-1}$.