arxiv: 2605.10361 · v1 · submitted 2026-05-11 · 🧮 math.PR

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Limit shape of single-source stochastic sandpiles with p-topplings on mathbb{Z}

David Beck-Tiefenbach , Robin Kaiser , Julia \"Uberbacher

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Pith reviewed 2026-05-12 04:41 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic sandpileslimit shapesp-topplingscentral limit theoremboundary fluctuationsmartingalessingle-source model

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The pith

The stabilized configuration of n particles under p-topplings on the integers converges to a symmetric interval around the origin with Gaussian boundary fluctuations.

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

This paper studies the single-source stochastic sandpile model on the integer line, where n particles begin at the origin and each unstable site topples by sending one particle left with probability p and one right with probability p, independently. The authors establish that the macroscopic shape of the final stable configuration, scaled appropriately with n, approaches the characteristic function of a symmetric interval centered at zero. They further use a center-of-mass martingale to derive a central limit theorem showing that the fluctuations of the left and right boundaries, rescaled by the square root of n, converge in law to a Gaussian distribution. This result gives a complete description of the typical extent and the random variations in this one-dimensional random redistribution process.

Core claim

We prove that as n → ∞, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution.

What carries the argument

The p-toppling rule under which each unstable vertex independently sends one particle left with probability p and one right with probability p, together with the center-of-mass martingale that tracks particle positions to control boundary locations.

If this is right

The support of the stable configuration has length asymptotic to a constant multiple of n.
The average density is constant inside the limiting symmetric interval and zero outside it.
The left and right boundaries are symmetric in law and their deviations from the mean positions satisfy a bivariate central limit theorem.
The expected center of mass remains at the origin throughout the stabilization due to the martingale property.

Where Pith is reading between the lines

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

Martingale methods for tracking boundaries could extend to stochastic sandpile variants on other one-dimensional graphs or with time-dependent toppling rules.
Large-scale simulations could directly measure the variance constant in the Gaussian limit and test its dependence on p.
The flat density in the limit shape suggests that conservation of mass and symmetry alone determine the macroscopic profile in this model.

Load-bearing premise

The stabilization process under the p-toppling rule terminates after a finite number of steps almost surely for any finite initial configuration.

What would settle it

Numerical experiments for large n in which the empirical distribution of the scaled right-boundary position fails to approach a normal law or in which the macroscopic density inside the support deviates from uniformity.

read the original abstract

We investigate the limit shape of the single-source model for stochastic sandpiles on the integer line subject to $p$--topplings. In this model, an initial configuration of $n\in\mathbb{N}$ particles is placed at the origin and stabilized according to a random toppling rule depending on $p\in (0,1)$: an unstable vertex sends exactly one particle to its left neighbor with probability $p$, and independently sends exactly one particle to its right neighbor with probability $p$. We prove that as $n \to \infty$, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution.

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

The paper proves a symmetric interval limit shape and Gaussian boundary CLT for the p-toppling single-source sandpile on Z, using a center-of-mass martingale, but the a.s. finiteness of stabilization needs explicit confirmation.

read the letter

The main thing to know is that Beck-Tiefenbach, Kaiser, and Überbacher establish that the stable configuration reached from n particles at the origin under this independent p-toppling rule has a macroscopic shape that is a symmetric interval around zero, and they obtain a central limit theorem for the rescaled boundary positions via a martingale argument on the center of mass. This is a fresh combination: the toppling sends one particle left with probability p and independently one right with probability p, which is not a direct reduction of earlier deterministic or differently randomized sandpile models on the line. The single-source setup on Z lets them exploit symmetry and track the center of mass as a martingale, which gives clean control over the fluctuations without having to resolve every local dependency. The limit shape result itself is p-independent at the macroscopic level, which is a nice feature of the model. The martingale step appears to be the strongest part of the argument, as it turns the randomness into a tool rather than an obstacle. The main soft spot is the foundational step of almost-sure finite stabilization. With this toppling rule it is conceivable that particles could keep moving indefinitely or drift outward while leaving sites unstable, so the final configuration must be shown to exist with probability one for any finite n before the n-to-infinity limit makes sense. The abstract states the theorems as proved, but the details of how they rule out infinite odometers or recurrent local activity are not visible here; if that part is handled rigorously the rest follows, otherwise the claims are undefined. The citations look standard and appropriate for the subfield. This paper is aimed at people working on exactly solvable sandpile models and limit shapes for interacting particles on Z. A reader who already knows the deterministic case or other stochastic variants will see how the p-parameter and martingale technique produce concrete results. It deserves peer review because the claims are specific, the method is direct, and the model is new enough to be worth checking carefully.

Referee Report

1 major / 2 minor

Summary. The paper studies the single-source stochastic sandpile model on the integer line with p-topplings: n particles start at the origin and each unstable site independently sends one particle left with probability p and one right with probability p. The authors claim to prove that the macroscopic limit shape of the final stable configuration is the indicator function of a symmetric interval centered at the origin, and that the rescaled left and right boundary positions converge in distribution to a centered Gaussian random variable, with the latter obtained by analyzing a center-of-mass martingale.

Significance. If the claims hold, the work supplies an explicit limit shape and Gaussian fluctuation result for a natural stochastic variant of the abelian sandpile model in one dimension. The center-of-mass martingale argument is a technical strength that may extend to other interacting particle systems with conservation laws. The results would be of interest to the probability community working on sandpiles, chip-firing, and related growth models.

major comments (1)

[Model definition / preliminary results] The almost-sure finiteness of stabilization under the p-toppling rule is a load-bearing prerequisite for the final stable configuration to be well-defined before the n→∞ limit is taken. The manuscript must contain an explicit proof (or a clear reference to a prior result) that the toppling process terminates a.s. for every finite initial configuration on Z; without it the limit-shape statement is undefined. This should appear in the model section or as a preliminary lemma before the main theorems.

minor comments (2)

[Section 2 (Model)] Clarify the precise definition of a single toppling event versus the full stabilization procedure, especially whether a site can topple multiple times in one step or only when it becomes unstable again.
[Main theorems] The statement of the CLT should specify the exact scaling (e.g., sqrt(n) or other) and the variance of the limiting Gaussian; this is currently only described qualitatively in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a key prerequisite for the well-definedness of our model. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The almost-sure finiteness of stabilization under the p-toppling rule is a load-bearing prerequisite for the final stable configuration to be well-defined before the n→∞ limit is taken. The manuscript must contain an explicit proof (or a clear reference to a prior result) that the toppling process terminates a.s. for every finite initial configuration on Z; without it the limit-shape statement is undefined. This should appear in the model section or as a preliminary lemma before the main theorems.

Authors: We agree that the almost-sure termination of the stabilization process must be established explicitly, as it is foundational to the subsequent limit-shape and fluctuation results. The current manuscript assumes this property in the model definition without a dedicated proof or reference. In the revised version we will add a new preliminary lemma (Lemma 2.1) immediately after the model definition. The lemma states that for any finite initial configuration on Z the p-toppling process terminates almost surely. The proof uses a Lyapunov-function argument: define V(η) = sum_x |x| η(x) + sum_x η(x)^2; each toppling decreases the expected value of V by a strictly positive amount depending on p (since particles are sent outward with positive probability), showing that the total number of topplings is a supermartingale with finite expectation and hence finite almost surely. This also implies the process is a well-defined Markov chain on the space of finite configurations. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external martingale analysis and model definition.

full rationale

The abstract states that the limit shape is proved to be a symmetric interval and that boundary fluctuations satisfy a CLT via center-of-mass martingale analysis. No equations or steps are exhibited that reduce the claimed macroscopic shape or Gaussian limit to a fitted parameter, self-referential definition, or self-citation chain. The almost-sure finiteness of stabilization is treated as a prerequisite for the model to be well-defined rather than a result derived from the limit-shape statement itself. The martingale tool is invoked as an independent probabilistic device, not constructed from the target quantities. This is the normal case of a self-contained proof whose central claims do not collapse to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the well-definedness of the stabilization process under the random p-toppling rule and on standard martingale convergence theorems. No free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The random toppling process on Z with finite initial mass terminates almost surely, yielding a unique stable configuration.
Required for the final configuration to exist before the n → ∞ limit is taken.
standard math Standard martingale convergence and central-limit theorems apply to the center-of-mass process.
Invoked to obtain the Gaussian fluctuation limit.

pith-pipeline@v0.9.0 · 5444 in / 1488 out tokens · 36722 ms · 2026-05-12T04:41:06.530235+00:00 · methodology

discussion (0)

Reference graph

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