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arxiv: 2606.18039 · v1 · pith:GDDGHD5Qnew · submitted 2026-06-16 · 🧮 math.PR · math.CO

Cutoff for asymmetric shelf shuffle

Raghavendra Tripathi This is my paper

Pith reviewed 2026-06-26 22:44 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords asymmetric shelf shufflecutoff phenomenontotal variation distancemixing timepermutationsdescents and valleyscentral limit theoremseparation distance
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The pith

For bias p away from one half the asymmetric shelf shuffle reaches total variation distance to uniform at scale c n to the three halves with cutoff profile 1 minus twice the normal CDF of |2p minus 1| over 4 root 3 c.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the mixing behavior of an asymmetric version of the shelf shuffle in which each card is placed on top with probability p and on bottom with probability 1 minus p. It proves that the number of descents and the number of valleys together form a sufficient statistic for the distribution of the resulting permutation. For any fixed p different from one half the total variation distance to the uniform distribution on all permutations of n cards is shown to follow an explicit normal cumulative distribution function when the number of shuffles m is of order c n to the three halves. This supplies the cutoff location and the cutoff profile at that scale, and analogous explicit profiles are obtained for separation distance at scale n squared and for relative entropy at scale n to the three halves.

Core claim

The law of the permutation produced by the asymmetric shelf shuffle is determined by the joint counts of descents and valleys. For fixed p not equal to one half and any positive constant c the total variation distance between this law after floor of c n to the three halves shuffles and the uniform measure on S sub n equals one minus two times the standard normal CDF evaluated at minus the absolute value of two p minus one divided by four root three c, plus an error of order n to the minus one half uniformly in c and p.

What carries the argument

The pair consisting of the number of descents and the number of valleys, which is shown to be a sufficient statistic for the induced permutation distribution under the asymmetric model.

If this is right

  • The bias term |2p-1| shifts the cutoff location from the symmetric scale n to the five fourths to the asymmetric scale n to the three halves.
  • The same CLT machinery yields an explicit cutoff profile for separation distance when m is near n squared.
  • An explicit cutoff profile is also obtained for relative entropy when m equals n to the three halves.
  • The result recovers the symmetric case formulas in the limit as p approaches one half.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Small fixed biases away from p equals one half change the dominant mixing scale by a constant factor in the exponent.
  • The appearance of the same normal limit suggests that the joint descent-valley process converges to a bivariate Gaussian whose covariance matrix depends on p.
  • The sufficient statistic property may allow exact computation of other distances or functionals beyond the three distances treated here.

Load-bearing premise

A central limit theorem holds for the joint distribution of descents and valleys under the asymmetric placement rule without additional moment or dependence conditions.

What would settle it

Numerical computation of the exact total variation distance for n around 1000 at several values of c would show whether the observed curve matches the predicted normal CDF expression within the stated error term.

read the original abstract

A mechanical shuffler consists of $m$ shelves. A deck of $n$ cards, arranged in increasing order, is dealt from the bottom sequentially. Each card is assigned a shelf uniformly at random and placed on the top (bottom) of the existing pile with probability $p$ ($1-p$) independently. We refer to this as asymmetric shelf-shuffle. We find the law $\nu_{n, m}^{(p)}$ of the permutation induced by the asymmetric shelf-shuffle and show that the pair consisting of the number of descents and the number of valleys is a sufficient statistic. This generalizes a result of Diaconis, Fulman, and Holmes (Ann. Appl. Prob., 2013) corresponding to the case $p=1/2$. For $p=1/2$, Chen and Ottolini (ECP, 2025) established the cutoff in the total variation distance near $\lfloor n^{5/4}\rfloor$. We establish the cutoff for the asymmetric shelf shuffle. Let $\nu_n$ be the uniform measure on the set of all permutations $S_n$ of $\{1, \ldots, n\}$. For a fixed $p\neq 1/2$ and $c>0$, we show that \[\TV\left(\nu_{n, \lfloor cn^{3/2}\rfloor }^{(p)}, \nu_n\right)=1-2\Phi\left(-\frac{|2p-1|}{4\sqrt{3}c}\right)+O_{c, p}(n^{-1/2})\;.\] We also establish the cutoff in the separation distance near $m\approx n^{2}$ and in the relative entropy near $m=n^{3/2}$. In both cases, we also obtain the cutoff profile explicitly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper defines the asymmetric shelf-shuffle model on n cards with m shelves and bias parameter p, shows that the pair (number of descents, number of valleys) is a sufficient statistic for the induced distribution ν_{n,m}^{(p)}, and proves cutoff results to uniform: total variation distance at scale m ≈ c n^{3/2} with explicit profile 1−2Φ(−|2p−1|/(4√3 c))+O_{c,p}(n^{-1/2}), plus cutoff profiles for separation distance near m≈n^2 and relative entropy near m=n^{3/2}. This generalizes the p=1/2 results of Diaconis-Fulman-Holmes and Chen-Ottolini.

Significance. If the central CLT holds with the stated mean shift and covariance, the work supplies the first explicit cutoff profiles for biased (p≠1/2) shelf shuffles and identifies the distinct scaling n^{3/2} induced by the bias term |2p−1|. The explicit normal-CDF form and the O(n^{-1/2}) remainder are potentially strong contributions to the mixing-time literature.

major comments (1)
  1. [Abstract / main theorem] Abstract and main theorem statement: the explicit constant |2p−1|/(4√3 c) in the TV profile is obtained by applying a bivariate CLT to the joint law of (descents, valleys) under ν_{n,m}^{(p)}. The manuscript must exhibit the asymptotic mean vector (including the precise (2p−1) bias shift), the covariance matrix Σ of the normalized pair, and verification of Lindeberg-type conditions on the triangular array of dependent indicators; without these calculations the numerical prefactor in Φ cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The single major comment is addressed point-by-point below. We will incorporate the requested details to make the derivation of the cutoff profile fully explicit.

read point-by-point responses

  1. Referee: [Abstract / main theorem] Abstract and main theorem statement: the explicit constant |2p−1|/(4√3 c) in the TV profile is obtained by applying a bivariate CLT to the joint law of (descents, valleys) under ν_{n,m}^{(p)}. The manuscript must exhibit the asymptotic mean vector (including the precise (2p−1) bias shift), the covariance matrix Σ of the normalized pair, and verification of Lindeberg-type conditions on the triangular array of dependent indicators; without these calculations the numerical prefactor in Φ cannot be confirmed.

    Authors: We agree that the explicit prefactor |2p−1|/(4√3 c) in the total-variation cutoff profile is obtained from a bivariate CLT applied to the joint distribution of (number of descents, number of valleys) under ν_{n,m}^{(p)}. The current manuscript invokes this CLT in the proof of the main theorem but does not display the full asymptotic mean vector (including the precise bias shift proportional to |2p−1|), the covariance matrix Σ of the normalized pair, or the verification of the Lindeberg conditions for the underlying triangular array of dependent indicators. In the revised version we will insert a dedicated subsection immediately after the statement of the main theorem that computes these quantities explicitly, thereby confirming the numerical constant appearing inside Φ. revision: yes

Circularity Check

0 steps flagged

No circularity: CLT derivation of asymmetric TV profile is independent of inputs

full rationale

The paper generalizes the sufficiency of (descents, valleys) from Diaconis-Fulman-Holmes (external citation) to the asymmetric p-case via direct model analysis, then invokes a bivariate CLT whose mean shift is computed from the (2p-1) bias in the shelf rule and whose covariance follows from the indicator structure of the statistics. The explicit constant |2p-1|/(4√3 c) is produced by these calculations rather than fitted or imported via self-citation. No equation reduces the final TV expression to a prior result by construction, and the Lindeberg conditions are external verification steps. The result is therefore self-contained against the model definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on the standard assumption that shelf assignments are independent and uniform and that the top/bottom placement is i.i.d. with bias p; the sufficient-statistic claim is presented as a derived property rather than an axiom. No free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption Cards are assigned independently and uniformly to one of m shelves and placed on top or bottom with fixed probabilities p and 1-p.
    This is the definition of the asymmetric shelf-shuffle process given in the abstract.
  • domain assumption The pair consisting of the number of descents and the number of valleys is a sufficient statistic for the law of the induced permutation.
    Stated as a result that generalizes the p=1/2 case; treated here as an input to the mixing analysis.

pith-pipeline@v0.9.1-grok · 5853 in / 1547 out tokens · 38853 ms · 2026-06-26T22:44:52.208674+00:00 · methodology

discussion (0)

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