Universality of Cutoff for Riffle Shuffling

TL;DR

This paper proves the cutoff phenomenon for riffle shuffles with general pile size distributions, extending previous results limited to binomial cuts, and characterizes the mixing time in terms of the distribution's limit.

Contribution

It establishes cutoff for riffle shuffles with arbitrary pile size distributions, including deterministic sequences, and extends to multi-partite shuffles, generalizing prior binomial-based results.

Findings

Cutoff occurs at a time proportional to log N with an explicit constant.

The 'cold spot' phenomenon determines the mixing time in general.

Results apply to both random and deterministic pile size sequences.

Abstract

A Gilbert-Shannon-Reeds (GSR) shuffle is performed on a deck of $N$ cards by cutting the top $n\sim Bin(N,1/2)$ cards and interleaving the two resulting piles uniformly at random. The celebrated "Seven shuffles suffice" theorem of [Bayer-Diaconis '92] established cutoff for this Markov chain: to leading order, total variation mixing occurs after precisely $\frac{3}{2}\log_2 N$ shuffles. Later work of [Lalley '00] and [Sellke '22] extended this result to asymmetric binomial cuts $n\sim Bin(N,p)$ for all $p\in (0,1)$. These results relied heavily on the binomial condition and many natural chains were left open, including uniformly random cuts and exact bisections. We establish cutoff for riffle shuffles with general pile size distribution. Namely, suppose the cut sizes $(n^{(t)})_{t\geq 1}$ are IID and the convergence in distribution $n^{(t)}/N \stackrel{d}{\to} \mu$ holds for some probability measure $\mu$ on the interval $[0,1]$. Then the mixing time $t_{mix}$ satisfies $t_{mix}/\log N\to \overline{C}_{\mu}$ for an explicit constant $\overline{C}_{\mu}$. The same result holds for any (deterministic or random) sequence of pile sizes with empirical distribution converging to $\mu$ on all macroscopic time intervals (of length $\Omega(\log N)$). It also extends to multi-partite shuffles where the deck is cut into more than $2$ piles in each step. Qualitatively, we find that the "cold spot" phenomenon identified by [Lalley '00] characterizes the mixing time of riffle shuffling in great generality.