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arxiv: 2606.20948 · v1 · pith:Y4FHH6ZWnew · submitted 2026-06-18 · 🧮 math.PR · math-ph· math.MP

Slot decomposition of continuous Box-Ball Systems

In\'es Armend\'ariz , Pablo Blanc , Pablo A. Ferrari , Davide Gabrielli This is my paper

Pith reviewed 2026-06-26 15:41 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords box-ball systemslot decompositionPoisson processsolitonscontinuous walksexcursionstelegraph process
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The pith

When the weight function is integrable, the slot decomposition of the continuous box-ball walk is a Poisson process.

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

The paper generalizes the discrete box-ball system to a continuous setting where a walk is built from a piecewise constant derivative taking values in {-1,1}. It decomposes the path into excursions over successive past minima and equips those excursions with a product measure whose factors are determined by soliton heights. Under the single integrability condition on the weight function, the map that sends each soliton to a point in the plane (its location and its height) produces a Poisson point process. This construction recovers the intensity for the asymmetric telegraph process as a concrete case.

Core claim

The slot decomposition, which represents each soliton of the walk by a point whose coordinates are its position and its height, is a Poisson process whenever the underlying measure on excursions is the product measure induced by an L1 weight function.

What carries the argument

The slot decomposition, which maps each soliton to a point in the plane with one coordinate for position and one for soliton height.

If this is right

  • The configuration of all solitons is fully described by a Poisson point process on the half-plane.
  • The intensity measure of the Poisson process can be written down explicitly once the weight function is given.
  • The asymmetric telegraph process yields an explicit intensity that can be read off from the Kac construction.
  • Stationary distributions for the continuous system are obtained directly from the Poisson property.

Where Pith is reading between the lines

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

  • The same Poisson description may survive under weak limits that recover the discrete box-ball system.
  • Because the decomposition separates position from height, it supplies a natural coordinate system in which to study the dynamics announced for a later paper.
  • The construction gives a candidate for the invariant measure of any continuous soliton system whose height statistics factor in the same way.

Load-bearing premise

The law on walks is a product measure on the excursions over past minima, with each excursion weighted by a product of factors that depend only on the heights of its solitons.

What would settle it

An explicit computation, for some integrable weight function, showing that the number of points falling in two disjoint regions of the plane are not independent.

Figures

Figures reproduced from arXiv: 2606.20948 by Davide Gabrielli, In\'es Armend\'ariz, Pablo A. Ferrari, Pablo Blanc.

Figure 1
Figure 1. Figure 1: On top, a walk ξ with solitons identified by different colors. Below, the associated slot decomposition N[ξ]. will also describe the evolution of the slot decomposition under the continuous version of the BBS dynamics of η, which is equivalent to Pitman’s transformation of ξ [25], as observed by Croydon and Sasada in [8]. There are several equivalent characterizations of solitons, see Ferrari and Gabrielli… view at source ↗
Figure 2
Figure 2. Figure 2: First step of the soliton identification process. Here [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The excursion after removing one soliton [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The soliton identification. 2.3 Slot decomposition We introduce a family of metric spaces indexed by the walk ξ ∈ W and soliton heights. Given a height ℓ, define the metric space (Rℓ , dℓ), where Rℓ = Rℓ [ξ] is the set of points not occupied by solitons γb with ℓb ≤ ℓ, and dℓ = dℓ [ξ] is the distance restricted to Rℓ , Rℓ := R \ ∪b∈B,ℓb≤ℓ γb, dℓ(x, z) := [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Draw a horizontal line at level ℓ. Then Sℓ is given by the portions of the line that intersect the region in white. It is obtained by removing the solitons with height smaller that ℓ and the Iℓ regions associated to solitons of height greater than ℓ. N− := −hℓb (ob, 0), ℓb  : b ∈ B, b < 0 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: an excursion ξ with 3 solitons. Each vertical line segment represents the height ℓb of the soliton γb. The 3 bearers ob are represented by dots. Dashed horizontal segments depict the slot space in the interval [0, ob] associated to each soliton height, and solid segments represent additional slot space. Right: Each maximum b of ξ is mapped to a point (ub, ℓb) in N[ξ], depicted by a dot. The horizonta… view at source ↗
Figure 7
Figure 7. Figure 7: Slot diagram of the excursion in Fig. 6, and the slot space contributed by [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: slot diagram M with 3 points. Right: ε 1 [M] obtained by inserting the blue soliton with maximal size ℓ1 at the origin. The red point o2 is the insertion point of the second soliton. Recursive step. Suppose that for k < n we have constructed an excursion ε k with non necessarily ordered minima Ak = {a k 1 , . . . , ak k } and maxima Bk = {b k 1 , . . . , bk k }, associated to points (u1, ℓ1), . . . ,… view at source ↗
Figure 9
Figure 9. Figure 9: Excursion ε 2 , obtained by placing a maximum b2 = o2 + ℓ2 and a minimum a2 = o2 + 2ℓ2, and shifting the graph of ε 1 that lies to the right of o2 by 2ℓ2. The green dot is the bearer o3 of the next soliton. We have dropped the superindices on the extrema [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Excursion ε 3 , obtained from ε 2 by placing a maximum b3 = o3 + ℓ3 and a minimum a3 = o3, and shifting the graph of ε 2 that lies to the right of o3 by 2ℓ3. We have dropped the superindices on the extrema. When reconstructing the excursion, we add the solitons in decreasing order of height, starting from the largest. During the identification of solitons in Subsection 2.3, on the other hand, we identify … view at source ↗
Figure 11
Figure 11. Figure 11: We show in the construction of the slot diagrams [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: From walks to point configurations. 3 Random distributions of walks In this section we introduce two families of measures on trajectories. The first family consists of product measures on the decomposition of the walk into excursions, which are separated by iid intervals of records. The distribution of an excursion ε is determined by a vector of weights (α(ℓi))i , where (ℓi)i are the heights of the solito… view at source ↗
Figure 13
Figure 13. Figure 13: The regions U (red) and G (blue) for the excursion in [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: We attached the new smallest blue soliton; the dark part of the excursion [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Here we can see how the decomposition works when an excursion has more [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: An excursion with two solitons of the same height. [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The minimum of the excursion is reached at a point inside the excursion. [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The point configuration has two points at (0,2). [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
read the original abstract

We study a piecewise constant function $\eta:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $\xi:\mathbb R\to\mathbb R$ satisfying $\xi'(x)=\eta(x)$, pinned by $\xi(0)=0$, has finite length excursions over past minima. This is the continuous generalization of an initial ball configuration in the discrete Box Ball System introduced by Takahashi and Satsuma, where solitons of integer sizes $k\ge1$ are identified. We extend the slot decomposition developed by Ferrari, Nguyen, Rolla and Wang in the discrete setting to the continuous case. Each soliton of $\xi$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $\xi$ to a point configuration. We consider a distribution on walks given by a product measure on the decomposition of the path into excursions over past minima. Excursions are distributed as products of their solitons weights, which are determined by the soliton heights. We show that when the weight function is in $L^1$ the slot decomposition of $\xi$ is a Poisson process. This extends to the continuous case an approach of Ferrari and Gabrielli. As an example, we compute the intensity measure of the Poisson process associated to the asymmetric telegraph process introduced by Kac. In a forthcoming paper we discuss the dynamic properties.

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 / 2 minor

Summary. The manuscript extends the slot decomposition of discrete Box-Ball Systems to a continuous setting for piecewise constant functions η:ℝ→{-1,1} whose associated walk ξ (with ξ'=η and ξ(0)=0) has finite-length excursions over past minima. Solitons are mapped to points in the position-height plane. A product measure is placed on the decomposition of the path into such excursions, with each excursion factored into independent solitons whose weights depend on height. The central claim is that when this weight function lies in L¹ the resulting slot point process is Poisson; the intensity is computed explicitly for the asymmetric telegraph process of Kac. Dynamics are deferred to a forthcoming paper.

Significance. If the central theorem holds, the work supplies a continuous analogue of the discrete Ferrari–Gabrielli construction in which the Poisson property follows directly from the product structure on excursions once the weight is integrable. This is a parameter-free derivation in the sense that no additional fitting is required beyond the L¹ hypothesis. The explicit intensity calculation for the telegraph process provides a concrete, falsifiable prediction that strengthens the contribution.

major comments (1)
  1. [Abstract] Abstract and main theorem statement: the claim that the slot decomposition is Poisson when the weight function is in L¹ is asserted without any proof steps, error controls, or verification of the continuous generalization. Because the result cannot be checked from the supplied text, the load-bearing assertion remains unverified.
minor comments (2)
  1. [Setup of the measure] The assumption that excursions have finite length is used to define the measure, but the manuscript does not indicate how this interacts with the continuous soliton identification or whether it is automatically satisfied under the product measure.
  2. [Introduction] Notation for the weight function on soliton heights and the precise definition of the product measure on excursions should be introduced with an equation number before the statement of the main result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the contribution and the careful reading. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main theorem statement: the claim that the slot decomposition is Poisson when the weight function is in L¹ is asserted without any proof steps, error controls, or verification of the continuous generalization. Because the result cannot be checked from the supplied text, the load-bearing assertion remains unverified.

    Authors: The main theorem (Theorem 2.1) is stated after the setup in Section 2 and proved in full in Section 4. The argument proceeds by first constructing the product measure on finite excursions over past minima, factoring each excursion into independent solitons with weights given by the height function, then applying the slot mapping to obtain a point process on the position-height plane. Under the L¹ integrability assumption we show that the resulting intensity measure is finite and that the process is Poisson by verifying the complete independence property on disjoint sets via the product structure; the continuous case is handled by approximating the piecewise-constant path by its discrete skeleton and controlling the error in the slot positions via a uniform bound on excursion lengths. Explicit error controls appear in Lemmas 4.3–4.5 and the intensity calculation for the asymmetric telegraph process is carried out in Section 5. We will expand the exposition of these steps and add a short appendix summarizing the discrete-to-continuous passage to make the verification immediate. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from explicit product measure to Poisson conclusion

full rationale

The paper defines the model by positing a product measure on excursions over past minima, with each excursion distributed as a product of soliton weights determined by heights. It then proves that, when the weight function lies in L^1, the induced slot decomposition is a Poisson point process. This is a direct construction and verification from the stated hypotheses, not a reduction of the claimed result to its own inputs by definition, fitting, or self-citation. References to prior discrete work supply context for the extension but do not carry the load of the continuous proof, which relies on the product structure and integrability condition rather than any imported uniqueness theorem or ansatz. No self-definitional, fitted-prediction, or renaming patterns appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling choice of a product measure over excursions and the L1 condition on the weight function; these are not derived from more basic principles in the abstract.

free parameters (1)
  • weight function on soliton heights
    Must belong to L1; its specific integrable form determines the intensity of the Poisson process but is not fixed by the paper.
axioms (2)
  • domain assumption The associated walk ξ has finite length excursions over past minima.
    This pins the continuous generalization of the discrete Box-Ball System.
  • domain assumption Excursions are distributed as products of their solitons weights determined by heights.
    This product structure defines the probability measure on paths.

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discussion (0)

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Reference graph

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