Mixing times via super-fast coupling

TL;DR

This paper introduces a novel coupling proof technique to analyze the mixing time of the transposition shuffle on a deck of n cards, achieving the known rate of Cn(log n) with a new probabilistic approach.

Contribution

It provides the first natural probabilistic coupling proof for the mixing rate of the transposition shuffle, expanding coupling methodology to include non-adapted rules.

Findings

Established a coupling proof matching the known mixing rate

Expanded coupling methodology to include non-adapted rules

Provided insights into probabilistic proofs for card shuffling

Abstract

We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate Cn(log{n}) with a moderate constant, C. This rate was determined by Diaconis and Shahshahani, but the question of a natural probabilistic coupling proof has been missing, and questions of its existence have been raised. The proof, and indeed any proof, requires that we enlarge the methodology of coupling to include intuitive but non-adapted coupling rules, because a typical Markovian coupling is incapable of resolving finer questions of rates.