Universality of Cutoff for Riffle Shuffling

TL;DR

This work proves a universal cutoff phenomenon for riffle shuffles under general pile-size distributions, showing that total-variation mixing occurs after a time proportional to log N with a constant dependent only on the cut-size distribution μ. The authors introduce the randomized-μ-like framework and derive explicit μ-dependent constants, including θ_μ, C_μ, tilde C_μ, and the universal time-scale constant overline{C}_μ, which governs the cutoff. The core methodology combines a robust shuffle-graph representation with a cold-spot (and countering hypotheses) perspective for the lower bound, and a truncated exponential-moment/Portenko-type upper-bound strategy, extended from multinomial to general μ-like and multi-partite shuffles. The results demonstrate that the cold-spot mechanism provides a versatile and sharp description of mixing not only for the classical binomial cuts but also for uniform and other natural pile-size distributions, establishing universality and several interesting consequences, including a non-convexity phenomenon and explicit numerical benchmarks for μ on the simplex.

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 $n^{(t)}/N \stackrel{d}{\to} μ$ holds for some probability measure $μ$ on the interval $[0,1]$. Then the mixing time $t_{mix}$ satisfies $t_{mix}/\log N\to \overline{C}_μ$ for an explicit constant $\overline{C}_μ$. The same result holds for any (deterministic or random) sequence of pile sizes with empirical distribution converging to $μ$ on all macroscopic time intervals (of length $Ω(\log N)$). 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.

Figures (1)

• Figure 1: As defined in Lemma \ref{['lem:another-expr']}, a matrix in $\widehat{\mathcal{M}}$ is obtained by taking a realization of $\overline{S}_K$ and then moving all rows lexicographically smaller than $s_i$ to the first $(i - y_{p,i} - \iota(p)+1)$ positions, followed by placing all remaining rows that start with $p$ in the next $n^{\mathop{\mathrm{\mathrm{rem}}}\nolimits}$ positions, while preserving their relative order. The portion enclosed by red bold lines corresponds exactly to a matrix in $\widehat{\mathcal{M}}$, and the four defining conditions of $\widehat{\mathcal{M}}$ are illustrated in the figure. The crucial Lemma \ref{['lem:cond-resam']}, which estimates the probability that all rows in the green part start with a given prefix $x$ conditioned on the blue part, is established by carefully counting the number of matrices whose green part satisfies certain prescribed properties.

Theorems & Definitions (115)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.3
• Proposition 1.4
• Theorem 2.1
• Proposition 2.2
• proof
• Lemma 2.3: hush2005concentration
• Lemma 2.4