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arxiv: 2605.19084 · v1 · pith:ULP3SYADnew · submitted 2026-05-18 · 🧮 math.PR · math.CO

Limit Profiles for Separation Distance

Peter E. Francis , Evita Nestoridi This is my paper

Pith reviewed 2026-05-20 07:08 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords separation distancelimit profilecutoff windowinverse riffle shufflerandom transpositionsspectral comparisonmixing time
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The pith

Separation distance limit profiles are determined for inverse riffle shuffles and random transpositions.

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

The paper focuses on the limiting behavior of separation distance to stationarity for two standard models of card shuffling as the deck size grows. Separation distance tracks the maximum probability gap from the uniform distribution over all starting points. The authors adapt spectral comparison methods and continuity arguments previously developed for total variation distance, showing these carry over to separation distance. This produces explicit limit profiles that capture the shape of the distance inside the cutoff window at times of the form t_n plus a constant times the window width w_n. The same framework is applied to random walks on product groups and the hypercube.

Core claim

We determine their separation distance limit profiles for inverse riffle shuffles and random transpositions. A limit profile records the limiting shape of the distance to stationarity inside the cutoff window, at times of the form t_n + c w_n. We develop a spectral comparison technique and study continuity properties in the style of prior works, adapted to separation distance. The comparison method is illustrated through random transpositions, as well as random walks on product groups and the hypercube.

What carries the argument

The spectral comparison technique adapted to separation distance, which transfers eigenvalue information and continuity properties to compute the explicit limit profiles inside the cutoff window.

If this is right

  • The separation distance cutoff occurs at the same leading time scale as total variation distance for inverse riffle shuffles and random transpositions.
  • Explicit formulas for the limit profiles give the precise rate at which the worst-case gap to uniformity shrinks or grows near the cutoff.
  • The adapted comparison technique applies directly to random walks on product groups and the hypercube.
  • Continuity of the profiles under small changes in the transition probabilities follows from the same arguments used for total variation.

Where Pith is reading between the lines

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

  • The technique may extend to other group-based Markov chains where the spectrum is known but separation distance analysis was previously unavailable.
  • These profiles could guide practical choices of how many shuffles are needed to reach a target closeness in applications like cryptography or simulations.
  • Comparing the shape of separation profiles with those for other distances such as chi-squared might reveal whether they share the same cutoff window or differ in their fine-scale behavior.

Load-bearing premise

The spectral comparison technique and continuity properties developed for total variation distance can be adapted to separation distance without introducing new obstructions.

What would settle it

Numerical computation of separation distance for finite but large decks at times t_n + c w_n, followed by checking whether the values converge to the predicted explicit profile as deck size tends to infinity.

read the original abstract

This paper studies limit profiles for the separation distance. A limit profile records the limiting shape of the distance to stationarity inside the cutoff window, at times of the form $t_n+cw_n$. We start with two famous card shuffles, a general setup for inverse riffle shuffles and random transpositions, and we determine their separation distance limit profiles. We then develop a spectral comparison technique and study continuity properties in the style of [Nes24; Nes25], adapted to separation distance. The comparison method is illustrated through random transpositions, as well as random walks on product groups and the hypercube.

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

2 major / 2 minor

Summary. The paper determines explicit limit profiles for the separation distance of inverse riffle shuffles and random transpositions inside the cutoff window. It develops a spectral comparison technique together with continuity properties adapted from prior total-variation work, and illustrates the method on random transpositions, random walks on product groups, and the hypercube.

Significance. If the adaptation is rigorous, the explicit profiles supply concrete benchmarks for two classical shuffles and the general comparison method could extend limit-profile results to separation distance on other chains. The work builds directly on the authors' earlier spectral techniques, so its value hinges on whether the new arguments control the pointwise minimum without extra factors.

major comments (2)
  1. [Section 3 (spectral comparison technique)] The central adaptation of spectral comparison from total variation to separation distance must control the minimizing state y inside the cutoff window. The manuscript asserts that the continuity properties carry over, but the argument for the min_y operation (which is not Lipschitz with respect to the spectral gap in the same manner as the integrated TV norm) is not fully detailed; a concrete bound or counter-example check would be needed to confirm that no additional logarithmic or oscillatory terms appear.
  2. [Section 2 (inverse riffle shuffles)] For the inverse riffle shuffle, the claimed separation limit profile is stated without an explicit derivation or error bound in the main text; the reduction to the prior total-variation profile appears to rely on a uniform control of the ratio P^t(x,y)/π(y) that is not verified pointwise.
minor comments (2)
  1. [Introduction] Notation for the separation distance s_n(t) should be introduced once with the precise definition max_x (1 - min_y P^t(x,y)/π(y)) rather than assumed from the abstract.
  2. [Section 3] The references to [Nes24; Nes25] should include a short sentence clarifying which continuity properties are reused verbatim and which require new justification for separation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications where possible and indicating the revisions we will make to strengthen the arguments and presentation.

read point-by-point responses

  1. Referee: [Section 3 (spectral comparison technique)] The central adaptation of spectral comparison from total variation to separation distance must control the minimizing state y inside the cutoff window. The manuscript asserts that the continuity properties carry over, but the argument for the min_y operation (which is not Lipschitz with respect to the spectral gap in the same manner as the integrated TV norm) is not fully detailed; a concrete bound or counter-example check would be needed to confirm that no additional logarithmic or oscillatory terms appear.

    Authors: We agree that the min_y operation requires explicit control to ensure the continuity properties adapt rigorously without extra factors inside the cutoff window. While the manuscript adapts the spectral comparison and continuity arguments from our prior total-variation results, the pointwise nature of the minimum does call for a dedicated bound. In the revision we will add a lemma in Section 3 establishing that |min_y (P^t(x,y)/π(y)) − min_y (Q^t(x,y)/π(y))| is bounded by a multiple of the spectral distance plus an o(1) term uniform in the window parameter c, thereby ruling out logarithmic or oscillatory contributions for the chains under consideration. A short verification for the hypercube example will also be included. revision: yes

  2. Referee: [Section 2 (inverse riffle shuffles)] For the inverse riffle shuffle, the claimed separation limit profile is stated without an explicit derivation or error bound in the main text; the reduction to the prior total-variation profile appears to rely on a uniform control of the ratio P^t(x,y)/π(y) that is not verified pointwise.

    Authors: The separation-distance limit profile follows from the explicit rising-sequence representation of the inverse riffle shuffle together with the known total-variation profile, using the relation sep(P^t,x) = max_y (1 − P^t(x,y)/π(y)). We acknowledge that the main text presents the reduction concisely and that a fully explicit pointwise error bound on the ratio inside the cutoff window is not written out. In the revised version we will expand the derivation in Section 2 to include the uniform control of the ratio and the resulting error term, confirming that the limit profile holds without additional factors. revision: yes

Circularity Check

1 steps flagged

Spectral comparison and continuity properties adapted from author's prior self-citations [Nes24; Nes25]

specific steps
  1. self citation load bearing [Abstract]
    "We then develop a spectral comparison technique and study continuity properties in the style of [Nes24; Nes25], adapted to separation distance."

    The central technique for controlling separation distance inside the cutoff window (via continuity properties) is developed by direct adaptation of prior results by co-author Nestoridi. If those prior results themselves rely on similar adaptations without external machine-checked or parameter-free benchmarks, the present claims on limit profiles inherit the same unverified chain rather than standing on independent derivation.

full rationale

The paper's core contribution is determining explicit separation distance limit profiles for inverse riffle shuffles and random transpositions via an adapted spectral comparison technique. This adaptation is explicitly styled after the author's own prior works, which creates moderate circularity risk for the load-bearing continuity properties when applied to the pointwise min_y operation in separation distance. However, the paper still contains independent content in applying the method to specific shuffles and product groups, so the derivation does not fully reduce to self-citation by construction. No self-definitional or fitted-input reductions are exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted; the central claims rest on unstated technical assumptions from the full proofs and prior works.

pith-pipeline@v0.9.0 · 5617 in / 974 out tokens · 34913 ms · 2026-05-20T07:08:38.171694+00:00 · methodology

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Reference graph

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