Cutoff for the cyclic adjacent transposition shuffle

TL;DR

This paper proves that the cyclic adjacent transposition (CAT) shuffle of n cards exhibits a cutoff at a specific mixing time, making it twice as fast as the standard adjacent transposition shuffle.

Contribution

It establishes the cutoff time for the CAT shuffle and compares its speed to the classic AT shuffle, showing it is significantly faster.

Findings

CAT shuffle exhibits cutoff at (n^3 / 2π^2) log n

CAT shuffle is twice as fast as the AT shuffle

Cutoff phenomenon confirmed for the systematic scan version

Abstract

We study the cyclic adjacent transposition (CAT) shuffle of $n$ cards, which is a systematic scan version of the random adjacent transposition (AT) card shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at $\frac{n^3}{2 \pi^2} \log n$, which concludes that it is twice as fast as the AT shuffle.