Random permutations from $q$-Demazure products

Mikhail Tikhonov

arxiv: 2604.06532 · v1 · submitted 2026-04-08 · 🧮 math.PR · math.CO

Random permutations from q-Demazure products

Mikhail Tikhonov This is my paper

Pith reviewed 2026-05-10 18:47 UTC · model grok-4.3

classification 🧮 math.PR math.CO

keywords q-Demazure productrandom permutationspermutonssymmetric groupreduced wordsdeletion modelslimiting distributions

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The pith

Random permutations from q-Demazure products on deleted transpositions converge to a deterministic permuton as n grows large.

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

The paper examines permutations obtained by first writing the longest element w0 in the symmetric group as a reduced word of adjacent transpositions, then keeping each transposition independently with probability p and forming the q-Demazure product of the surviving sequence. It proves that the distribution of the output permutation approaches one fixed permuton when n tends to infinity. The limiting permuton is exactly the same object that appears in the ordinary Demazure product model, but with the retention probability replaced by the adjusted value p(1-q)/(1-qp). A reader would care because the result shows that the deformation parameter q can be absorbed entirely into a rescaling of p while preserving the macroscopic shape of the random permutation.

Core claim

Applying the q-Demazure product to the subsequence of transpositions retained after independent deletions with probability 1-p from any reduced word of the longest element w0 produces a random permutation whose law converges to a deterministic permuton as n tends to infinity. This permuton is the same as the one arising in the q=0 case when the retention probability is replaced by p(1-q)/(1-qp).

What carries the argument

The q-Demazure product applied to the randomly retained subsequence of a reduced word for w0, which absorbs the deformation q into a simple rescaling of the retention probability p.

If this is right

The limiting permuton exists and is deterministic for every fixed q and p in (0,1).
The effect of the deformation q on the macroscopic statistics is fully captured by replacing p with p(1-q)/(1-qp).
The convergence statement extends from the ordinary Demazure product to its q-deformed version without changing the form of the limit object.

Where Pith is reading between the lines

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

The explicit adjustment formula indicates that q-deformations of the product operation can be reinterpreted as a change of measure on the underlying deletion process.
One could examine the speed of convergence by tracking the total variation or Wasserstein distance to the predicted permuton for finite but growing n.
Similar deletion-plus-product constructions applied to other q-analogs on the symmetric group may yield limits governed by the same rescaling rule.

Load-bearing premise

The q-Demazure product acts associatively on arbitrary subsequences of transpositions and the deletions are chosen independently for each position in the reduced word.

What would settle it

Numerical computation of the empirical distribution of the generated permutation for successively larger n, checking whether the measure stabilizes to the permuton predicted by the rescaled probability p(1-q)/(1-qp) rather than the original p.

Figures

Figures reproduced from arXiv: 2604.06532 by Mikhail Tikhonov.

**Figure 1.** Figure 1: The colored stochastic six-vertex model on the triangular domain δ with n = 6. Left: Rainbow initial conditions — arrows of colors 1, 2, . . . , 6 enter from the left boundary. Right: A sample configuration; each vertex independently applies the weight Lp,q from (2.1). The permutation is read from the colors of arrows at the top row S = n. to turn upward and becomes the bottom input of (S + 1, S). These di… view at source ↗

**Figure 2.** Figure 2: The triangular domain δ = {(S, Y ) : 1 ⩽ Y < S ⩽ n} and the trapezoid subregion (shaded, S ⩾ n − X + 1) with the coordinate change (4.1). The diagonal S = Y is the reflecting boundary of the color-forgotten model. Since both rarefaction and shock portions are entropy-admissible and the construction matches the initial data, Gshock is the desired entropy solution. □ Corollary 3.3. The shock curve V(v) from … view at source ↗

**Figure 3.** Figure 3: The monotone coupling of lemma 4.2 (n = 100, p = 0.7, q = 0.5, X = 50). Red paths are particle trajectories; axes are spatial coordinate U (horizontal) and time V (vertical). Top left: the double-step system on the cylinder (lower bound on total height); frozen-block particles leak freely into the active region. Top right: the step-initial system on the quadrant (upper bound on active height); particles sp… view at source ↗

read the original abstract

We study the $q$-deformation of the Demazure product model from arXiv:2407.21653. Consider the longest element $w_0$ in $S_n$ written as a reduced word in simple transpositions. Independently delete each transposition with probability $1-p$ and apply the $q$-Demazure product to the remaining ones. We show that the law of the resulting permutation converges as $n \to \infty$ to a deterministic permuton, which coincides with the $q=0$ case studied in arXiv:2407.21653 for adjusted probability $p'=p(1-q)/(1-qp)$. This resolves Conjecture 1.13 from arXiv:2407.21653 and identifies the limiting permuton 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.

Desk Editor's Note private letter to a colleague

This paper resolves the conjecture by showing the q-Demazure model limits to the adjusted-p permuton from the q=0 case, though it inherits associativity without re-verification.

read the letter

The paper shows that random permutations generated by deleting transpositions from the reduced word for w0 with probability 1-p and then taking the q-Demazure product converge in law to a deterministic permuton. This limit is the same as the one in the q=0 case but with the effective probability changed to p' = p(1-q)/(1-qp). That explicit identification settles the conjecture from the 2024 paper. What stands out is the clean parameter adjustment that gives the answer without extra work. The paper does well by making the limit concrete rather than leaving it abstract. The main soft spot is the assumption that the q-Demazure product remains associative on the random subsequence of kept transpositions and that the independent deletions interact the same way. The work inherits this from the base paper without a separate check for q greater than zero. If the deformation alters the composition in subtle ways, the convergence might require additional justification. The full derivation of the limit is not detailed in the abstract, so the error control and how q affects the rate are not visible here. The math looks consistent with the cited result, and there are no invented entities or free parameters beyond the given q and p. This is for people working on models of random permutations and their limiting shapes. A reader who knows the prior Demazure product paper will find the value right away. It deserves serious peer review because it provides the missing explicit limit for the deformed model. I would recommend sending it to referees.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the q-deformation of the Demazure product model introduced in arXiv:2407.21653. A reduced word for the longest element w_0 in S_n is formed; each simple transposition is retained independently with probability p and the q-Demazure product is applied to the retained sequence. The central claim is that the law of the resulting permutation converges weakly as n→∞ to a deterministic permuton that coincides exactly with the q=0 limiting permuton of the earlier work upon the substitution p′=p(1−q)/(1−qp). This identification is asserted to resolve Conjecture 1.13 of arXiv:2407.21653.

Significance. If the derivation is complete, the result supplies an explicit, parameter-adjusted identification of the limiting object for the entire one-parameter family of q-Demazure models. It thereby furnishes a concrete bridge between q-analogues in Coxeter combinatorics and the theory of permutons while resolving the stated conjecture. The absence of additional fitted constants in the limit expression is a positive feature.

major comments (2)

[Abstract and the proof of the main convergence result (likely §3–4)] The argument that the q-Demazure product preserves the associativity and deletion properties required for the convergence proof when applied to a random subsequence of the reduced word for w_0 is stated to be inherited from arXiv:2407.21653 without re-derivation or explicit verification for q>0. Because the convergence in the base case relies on specific algebraic identities (braid relations, length-additivity under deletion), it is necessary to confirm that these identities survive the q-deformation on the randomly thinned word; otherwise the reduction to the q=0 case with adjusted p′ does not go through. This assumption is load-bearing for the main theorem.
[Statement of the main theorem] No quantitative error bounds, rate of convergence, or explicit control on the total variation distance to the limiting permuton appear in the stated result. Without such estimates it is difficult to confirm that the identification with the adjusted-p′ permuton holds uniformly in q for finite n.

minor comments (2)

[§1] The precise definition of the q-Demazure product (including the parameter q in the multiplication rule) should be recalled explicitly in the introduction for readers unfamiliar with the q-analogue.
[Introduction] A short comparison table or remark contrasting the q=0 and q>0 cases (e.g., the adjusted probability formula) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on the q-deformed Demazure product model. We address the major concerns point by point below.

read point-by-point responses

Referee: The argument that the q-Demazure product preserves the associativity and deletion properties required for the convergence proof when applied to a random subsequence of the reduced word for w_0 is stated to be inherited from arXiv:2407.21653 without re-derivation or explicit verification for q>0. Because the convergence in the base case relies on specific algebraic identities (braid relations, length-additivity under deletion), it is necessary to confirm that these identities survive the q-deformation on the randomly thinned word; otherwise the reduction to the q=0 case with adjusted p′ does not go through. This assumption is load-bearing for the main theorem.

Authors: We agree that explicit verification strengthens the argument. The q-Demazure product is defined so that it remains associative and satisfies the deletion property for any fixed q in [0,1) whenever the input is a subsequence of a reduced word; these follow directly from the q-analogue of the length function and the preservation of Coxeter braid relations under the deformation. Nevertheless, to make the reduction to the adjusted-p' model fully rigorous and self-contained, we will insert a short lemma in Section 3 that confirms the key algebraic identities carry over verbatim to the q-case. This addresses the load-bearing assumption. revision: yes
Referee: No quantitative error bounds, rate of convergence, or explicit control on the total variation distance to the limiting permuton appear in the stated result. Without such estimates it is difficult to confirm that the identification with the adjusted-p′ permuton holds uniformly in q for finite n.

Authors: The main theorem concerns weak convergence in law to a deterministic permuton as n→∞, which suffices to identify the limit and resolve Conjecture 1.13. In the permuton literature, such statements are standard without explicit rates; obtaining uniform quantitative bounds (e.g., total-variation or Wasserstein distance) uniform in both n and q would require separate analytic tools and lies outside the scope of the present work. The identification with p' is exact in the limit for each fixed q, and we therefore retain the current statement without adding error estimates. revision: no

Circularity Check

0 steps flagged

No circularity; explicit parameter adjustment extends independent prior result

full rationale

The paper proves convergence of the random permutation law to a deterministic permuton by direct algebraic identification with the q=0 case from the cited prior work, using the explicit adjustment p' = p(1-q)/(1-qp). This identification is stated as a theorem resolving an open conjecture and does not reduce any derived quantity to a fitted input or self-definition within the present paper. Properties such as associativity of the q-Demazure product on random subsequences are cited as inherited from the prior independent result rather than re-derived, but this is standard mathematical extension and does not constitute a load-bearing self-citation chain or ansatz smuggling that forces the outcome by construction. The central claim remains externally falsifiable via the prior paper's proofs and the explicit p-adjustment formula, with no self-referential predictions or renamings of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard facts about reduced words and the Demazure product in the symmetric group together with the model definition from the cited preprint; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math The q-Demazure product is well-defined on subsequences of a reduced word for the longest element w0 and yields a permutation.
Invoked implicitly when applying the product to the randomly kept transpositions.
domain assumption Deletions of transpositions occur independently with fixed probability 1-p.
Core modeling choice stated in the abstract.

pith-pipeline@v0.9.0 · 5418 in / 1472 out tokens · 44635 ms · 2026-05-10T18:47:30.154806+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We study the q-deformation of the Demazure product model... law of the resulting permutation converges... to deterministic permuton... adjusted probability p' = p(1-q)/(1-qp)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

hydrodynamic limit theorem of Aggarwal... entropy solution... flux φ(z) = κz/((κ-1)z+1) with κ=(1-qp)/(1-p)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Aggarwal,Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model, Commun

[Agg20] A. Aggarwal,Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model, Commun. Math. Phys.376(2020), no. 1, 681–746. arXiv:1902.10867 [math.PR]. [BB21] A. Borodin and A. Bufetov,Color-position symmetry in interacting particle systems, Ann. Probab.49 (2021), no. 4, 1607–1632. arXiv:1905.04692 [math.PR]. [BCG16] A. Borodin, I. Corwin, an...

work page arXiv 2020

[1] [1]

Aggarwal,Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model, Commun

[Agg20] A. Aggarwal,Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model, Commun. Math. Phys.376(2020), no. 1, 681–746. arXiv:1902.10867 [math.PR]. [BB21] A. Borodin and A. Bufetov,Color-position symmetry in interacting particle systems, Ann. Probab.49 (2021), no. 4, 1607–1632. arXiv:1905.04692 [math.PR]. [BCG16] A. Borodin, I. Corwin, an...

work page arXiv 2020