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arxiv: 2606.12670 · v1 · pith:7DWQO3WM · submitted 2026-06-10 · math.PR · math-ph· math.CO· math.MP

The censored stochastic six-vertex model and parabolic Kazhdan--Lusztig R-polynomials

Hindy Drillick , Levi Haunschmid-Sibitz This is my paper

Reviewed by Pith2026-06-27 08:06 UTCgrok-4.3pith:7DWQO3WMopen to challenge →

classification math.PR math-phmath.COmath.MP
keywords stochastic six-vertex modelcensored processstochastic dominationparabolic Kazhdan-Lusztig R-polynomialsIwahori-Hecke algebrasecond-class particlesintertwining relation
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The pith

The censored stochastic six-vertex model started from the step initial condition is stochastically dominated by the blocking measure when b1 < b2.

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

The paper introduces a censored version of the stochastic six-vertex model. For parameters b1 less than b2, the model begun from the initial condition 1_{x>0} is shown to be stochastically dominated at every time by the blocking measure. This supplies a partial analog of the censoring inequality for monotone spin systems and permits control of second-class particles. The demonstration proceeds via parabolic Kazhdan-Lusztig R-polynomials that arise from a connection to the Iwahori-Hecke algebra of the symmetric group. An intertwining relation is also obtained with the normalized R-polynomials acting as kernel.

Core claim

We introduce a censored version of the stochastic six-vertex model. We show that for parameters b_1 < b_2, this model started from the initial condition 1_{x>0} is stochastically dominated at any time by the blocking measure. This is a partial analog of the censoring inequality for monotone spin systems. In particular, this result allows us to control the behavior of second-class particles. The proof uses parabolic Kazhdan--Lusztig R-polynomials, whose appearance is explained using a connection between the stochastic six-vertex model and the Iwahori--Hecke algebras of symmetric groups. Furthermore, we find an intertwining relation for this process using normalized parabolic Kazhdan--Lusztig

What carries the argument

The censored stochastic six-vertex model, whose censoring operation is chosen to preserve a direct link to the Iwahori-Hecke algebra so that parabolic Kazhdan-Lusztig R-polynomials can be used to prove stochastic domination by the blocking measure.

If this is right

  • The domination result controls the behavior of second-class particles.
  • The construction supplies a partial analog of the censoring inequality known for monotone spin systems.
  • An intertwining relation holds with normalized parabolic Kazhdan-Lusztig R-polynomials serving as the kernel.
  • The algebraic connection to the Iwahori-Hecke algebra accounts for the appearance of the R-polynomials in the proof.

Where Pith is reading between the lines

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

  • The same censoring construction might extend to other integrable models that admit an Hecke-algebra representation.
  • Numerical checks on finite segments could test whether the domination holds for small times and moderate lattice sizes.
  • The intertwining kernel may yield explicit moment formulas or generating functions for the censored process.
  • The domination could be used to bound the speed or fluctuations of second-class particles in the uncensored six-vertex model as well.

Load-bearing premise

The specific definition of the censoring operation preserves the monotonicity and the algebraic connection to the Iwahori-Hecke algebra of the symmetric group that is needed for the R-polynomials to control the domination.

What would settle it

A direct computation or simulation for concrete b1 < b2 showing that, at some positive time, the censored process begun from 1_{x>0} produces a state with more particles than the blocking measure would falsify the claimed stochastic domination.

Figures

Figures reproduced from arXiv: 2606.12670 by Hindy Drillick, Levi Haunschmid-Sibitz.

Figure 1
Figure 1. Figure 1: The six allowed configurations for the stochastic six-vertex model. 1.2. The model. We define the stochastic six-vertex model as a measure on collections of up-right paths in the upper half plane Z × Z≥0. We interpret the first coordinate of vertices (x, t) ∈ Z × Z≥0 as the spatial coordinate and the second coordinate as time. All paths start on the horizontal line {(x, 0) : x ∈ Z}. Paths are allowed to sh… view at source ↗
Figure 2
Figure 2. Figure 2: A piece of Z×Z≥0 where some of the vertices have been censored. system and running a censored Glauber dynamics where the set of transitions censored for ν˜ contains the set censored for ν. Then we have that (2) ˜ν ⪯ ν ⪯ π , where π is the stationary measure of the Glauber chain. It is not surprising that the full version of this theorem does not hold for the stochastic six-vertex model, since the stochasti… view at source ↗
Figure 3
Figure 3. Figure 3: The edges whose occupation variables are the ηt,k are shown as arrows. Remark 2.3. Note that the Pk act on Dirac measures in the following way: (3) Pkδη =    (1 − b1)δη + b1δη k , if (η(k), η(k + 1)) = (0, 1) (1 − b2)δη + b2δη k if (η(k), η(k + 1)) = (1, 0) δη, if η(k) = η(k + 1), as can be easily checked from the definition. Applying these operators in the correct order to the initial condition gives… view at source ↗
Figure 4
Figure 4. Figure 4: The rim decomposition of the partition (5, 5, 3, 2, 1), where each color represents a rim. Definition 3.6 (Parabolic Kazhdan–Lusztig R-measures). For a partition λ with rim￾decomposition (λ (k) )k≥1 and a partition µ ≤ λ, define (7) vλ(µ) := q |µ| Y k≥1 (1 − q k ) nk , where nk is the number of connected components of the partition λ (k) (or equivalently of the rim λ (k)/λ(k+1)) when all boxes in µ are rem… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of how to compute vλ(µ) for λ = (5, 5, 3, 2, 1) and different choices of µ. On the left we have µ = (3, 3) and vλ(µ) = q 6 (1−q) 2 (1− q 2 ) 2 , while on the right we have µ = (3, 1, 1) and vλ(µ) = q 5 (1 − q)(1 − q 2 ) 2 . The functions vλ, which we see as vectors in the vector space spanned by formal linear combinations of partitions, have the following alternative description which will … view at source ↗
Figure 6
Figure 6. Figure 6: The q-labeling of the partition (5, 5, 3, 2, 1). Lemma 3.9 (q-labeling description of vλ). The function vλ has the following alternative description. (8) vλ(µ) = q |µ| Y (i,j)∈λ/µ wλ(i, j). Proof. In the ribbon λ (k)/λ(k+1), the boxes not filled with 1 are filled with either (1 − q k ) or (1 − q k ) −1 . The values (1 − q k ) are placed in exactly the corners where the ribbon turns left (when traversing th… view at source ↗
Figure 7
Figure 7. Figure 7: The allowed configurations for the multi-class stochastic six-vertex model, where red lines represent class i and blue lines represent class j for i < j. Proof. Since all vertices on the boundary {(k + 1 + t, t) : t ≥ 0} are censored, ηt(x) = 1 for all x ≥ k + 1. Therefore, it does not change the process if we censor all vertices to the right of the boundary as well, i.e., all vertices {(x, t) : x ≥ k + 1 … view at source ↗
read the original abstract

We introduce a censored version of the stochastic six-vertex model. We show that for parameters $b_1 < b_2$, this model started from the initial condition ${1}_{x>0}$ is stochastically dominated at any time by the blocking measure. This is a partial analog of the censoring inequality for monotone spin systems. In particular, this result allows us to control the behavior of second-class particles. The proof uses parabolic Kazhdan--Lusztig $R$-polynomials, whose appearance is explained using a connection between the stochastic six-vertex model and the Iwahori--Hecke algebras of symmetric groups. Furthermore, we find an intertwining relation for this process using normalized parabolic Kazhdan--Lusztig $R$-polynomials as an intertwining kernel.

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 / 1 minor

Summary. The paper introduces a censored version of the stochastic six-vertex model. It proves that, for parameters b1 < b2, the process started from the initial condition 1_{x>0} is stochastically dominated at every time by the blocking measure. The argument relies on a connection to Iwahori-Hecke algebras of symmetric groups that lets parabolic Kazhdan-Lusztig R-polynomials control the domination; the same polynomials are used to establish an intertwining relation for the censored process.

Significance. If the domination holds, the result supplies a partial analog of the classical censoring inequality for monotone spin systems inside an integrable model, yielding control on second-class particles. The explicit use of parabolic KL R-polynomials as both domination bounds and intertwining kernels is a concrete algebraic-probabilistic link that is not common in the six-vertex literature.

major comments (1)
  1. [Abstract (proof-method paragraph)] The central claim rests on the assertion that the censoring operation preserves both monotonicity and the precise algebraic connection to the Iwahori-Hecke algebra needed for the R-polynomials to dominate the transition kernel. No explicit verification of this preservation (e.g., how the censored rates modify the generators or the Hecke relations) appears in the outline given in the abstract or the proof-method paragraph.
minor comments (1)
  1. [Introduction] The blocking measure and the precise form of the initial condition 1_{x>0} should be recalled with a short display equation in the introduction for readers who do not have the six-vertex literature at hand.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful comment on the abstract. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract (proof-method paragraph)] The central claim rests on the assertion that the censoring operation preserves both monotonicity and the precise algebraic connection to the Iwahori-Hecke algebra needed for the R-polynomials to dominate the transition kernel. No explicit verification of this preservation (e.g., how the censored rates modify the generators or the Hecke relations) appears in the outline given in the abstract or the proof-method paragraph.

    Authors: We agree that the abstract's proof-method paragraph would benefit from a brief indication of how censoring preserves monotonicity and the algebraic link to the Iwahori-Hecke algebra. The detailed verification (including the effect of the censored rates on the generators and the preservation of the Hecke relations) is given in the main text, but we will revise the abstract to include a short clarifying sentence on this point, with a pointer to the relevant sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result (stochastic domination of the censored six-vertex model by the blocking measure) is derived via an external algebraic connection to Iwahori-Hecke algebras and parabolic Kazhdan-Lusztig R-polynomials. This connection is invoked as an independent structure rather than being defined in terms of the target domination statement or fitted from the model's outputs. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described proof method. The censoring operation is presented as preserving monotonicity and the algebraic link, but this is an assumption on the definition, not a reduction of the result to itself by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces one new model variant and relies on standard background facts from probability and algebra; no free parameters or invented physical entities are visible in the abstract.

axioms (2)
  • domain assumption Stochastic domination is preserved under the censoring operation for the chosen parameters
    Invoked to obtain the partial analog of the censoring inequality.
  • domain assumption The stochastic six-vertex model admits a representation via Iwahori-Hecke algebras of symmetric groups
    Explains the appearance of the parabolic Kazhdan-Lusztig R-polynomials.
invented entities (1)
  • Censored stochastic six-vertex model no independent evidence
    purpose: To obtain stochastic domination by the blocking measure and control of second-class particles
    New variant introduced in the paper; no independent evidence outside the construction is given.

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