Partial mixing of semi-random transposition shuffles

TL;DR

This paper investigates the mixing times of various semi-random transposition shuffles, establishing bounds and cutoff phenomena for subsets of cards, using coupling methods to relate the mixing of multiple cards to single-card mixing.

Contribution

It provides new bounds and cutoff results for the mixing times of subsets of cards in semi-random transposition shuffles, extending understanding of partial mixing in these processes.

Findings

Mixing time for any $k$ cards is at most $n ext{log}k$ for semi-random transposition shuffles.

Cutoff occurs at this time for top-to-random shuffle when $k o\infty$, with a window of size $O(n)$.

Cutoff at time $(1/2)n ext{log}k$ for random-to-random transposition shuffle.

Abstract

We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is cutoff at this time with a window of size O(n), provided further that $k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$ for the same conditions on $k$. Finally, we analyse the cyclic-to-random transposition shuffle and show partial mixing occurs at time $\le\alpha n\log k$ for some $\alpha$ just larger than 1/2. We prove these results by relating the mixing time of $k$ cards to the mixing of one card. Our results rely heavily on coupling arguments to bound the total variation distance.