Pool model: a mass preserving multi particle aggregation process

Eviatar B. Procaccia; Yuan Zhang; Zhenhao Cai

arxiv: 2604.14851 · v1 · submitted 2026-04-16 · 🧮 math.PR · math-ph· math.MP

Pool model: a mass preserving multi particle aggregation process

Zhenhao Cai , Eviatar B. Procaccia , Yuan Zhang This is my paper

Pith reviewed 2026-05-10 10:12 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords Pool modelmulti-particle aggregationdiffusion-limited aggregationKurtz theoremPoisson point processmass conservationrandom walksrotational symmetry

0 comments

The pith

The Pool model grows a circular pool by absorbing random-walking particles while exactly preserving mass through proportional area expansion.

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

The paper defines the Pool model in the plane as a rotationally symmetric version of multi-particle diffusion-limited aggregation. Particles execute continuous-time random walks and are absorbed upon reaching the boundary of a central circular pool, whose radius then increases so that the added area matches the absorbed mass. The central result is an adapted version of Kurtz's theorem: conditioned on the pool's growth path, the positions of all particles form an independent non-homogeneous Poisson point process. This construction yields a mass-preserving dynamics whose particle field admits an exact probabilistic description. A reader would care because the model replaces the irregular boundary of standard aggregation with a tractable circular interface while retaining the essential absorption and growth mechanism.

Core claim

The Pool model is introduced as a mass-preserving process in which a circular pool centered at the origin absorbs particles performing continuous-time random walks; each absorption increases the pool's area by a fixed amount corresponding to the particle mass. The authors establish a version of Kurtz's theorem showing that the configuration of particles, conditioned on the realized growth of the pool, is distributed as an independent non-homogeneous Poisson point process whose intensity depends on the pool's radius history.

What carries the argument

The adapted Kurtz theorem, which represents the particle field conditioned on pool growth as an independent non-homogeneous Poisson point process.

If this is right

The Poisson representation supplies an exact sampling method for the particle configuration given any pool-growth path.
Mass is conserved exactly because area expansion is set equal to the number of absorbed particles.
The model supplies a rotationally symmetric benchmark against which asymmetric aggregation processes can be compared.
Growth rates and interface fluctuations can be studied directly through the intensity function of the Poisson process.

Where Pith is reading between the lines

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

The same conditioning argument might extend to growth laws other than area-proportional expansion, such as radius-proportional or curvature-driven rules.
Numerical simulations could test how quickly deviations from circularity appear when the rotational symmetry is broken by initial conditions or noise.
The Poisson description could be used to derive large-scale hydrodynamic limits or density equations without tracking individual particles.

Load-bearing premise

The pool is forced to remain perfectly circular and the whole dynamics are required to stay rotationally symmetric at every time.

What would settle it

Compute the empirical point pattern of particles at a fixed time, condition on the observed pool radius path, and test whether the pattern deviates from the intensity measure of the predicted non-homogeneous Poisson process.

Figures

Figures reproduced from arXiv: 2604.14851 by Eviatar B. Procaccia, Yuan Zhang, Zhenhao Cai.

**Figure 1.** Figure 1: Simulations on the growth of Et , t ∈ [0, 100] in an approximated engulf pool model, where 1) free particles move in a finite box of size 800 × 800 with periodic boundary conditions; and 2) engulfing is done at the end of each small deterministic time step δt = 10−2 . 20 random realizations are recorded. (2) If one replaces in the model the continuous time and space random walks with Brownian motion, our … view at source ↗

read the original abstract

We present and study the Pool model in $\mathbb{R}^2$, a rotationally symmetric analogue of Multi-Particle Diffusion-Limited Aggregation (MDLA), in which particles ("droplets") perform continuous-time random walks and are absorbed upon entering a circular pool initially centered at the origin. Each absorbed particle increases the pool's mass, and the pool expands so that its area grows accordingly, yielding a natural mass-preserving dynamics. A central tool which is of independent interest is a version of Kurtz's theorem for this model, depicting the field of particles conditioned on the growth of the pool as an independent non-homogeneous Poisson point process.

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 Pool model adds a clean, mass-preserving symmetric version of MDLA plus a Kurtz-type theorem that follows from standard Poisson thinning under the imposed circularity.

read the letter

The paper introduces the Pool model in R^2 as a rotationally symmetric analogue of multi-particle diffusion-limited aggregation. Particles perform independent continuous-time random walks and get absorbed into a growing circular pool whose radius increases deterministically with absorbed mass to keep area proportional to total mass. The central result is a version of Kurtz's theorem: conditioned on the pool growth path, the particle field is an independent non-homogeneous Poisson point process. This follows directly from the independence of the walks and the thinning property once the radius function is fixed by the conditioning. The stress-test note confirms the step has no hidden dependence or inconsistency under the model's rules. That is the main new piece and it is stated clearly from first principles without reducing to fitted parameters or prior published results on MDLA. The construction is explicit and the representation is a usable tool for anyone who wants to study conditioned dynamics in this setting. The paper does well by keeping the mass conservation built into the radius evolution rather than imposed afterward. The symmetry makes the Poisson representation work without extra work. The main limitation is exactly that symmetry: the pool is forced to stay circular and the dynamics rotationally invariant at every time. This is a modeling choice that simplifies the analysis but means the results do not directly address asymmetric growth that appears in more general aggregation. The abstract gives no proof details or error bounds, so the full paper needs to show the thinning argument is applied without gaps, but nothing in the stated claim looks unsupported. This work is for researchers in mathematical probability who study interacting particle systems and aggregation models. A reader looking for explicit representations or Kurtz-type tools in simplified symmetric cases will get something concrete to use or extend. It deserves a serious referee because the model is new, the theorem rests on standard facts applied cleanly, and the limitation is stated up front rather than hidden. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Pool model in R², a rotationally symmetric analogue of multi-particle diffusion-limited aggregation. Particles perform independent continuous-time random walks and are absorbed at the boundary of a circular pool centered at the origin; each absorption increases the pool mass, and the radius is updated so that the area grows proportionally to preserve mass. The central result is a version of Kurtz's theorem asserting that the particle configuration, conditioned on the (deterministic) pool-growth path, is distributed as an independent non-homogeneous Poisson point process.

Significance. If the theorem holds, the Poisson representation supplies a tractable description of the particle field that decouples positions from the shared growth process. This is of independent interest for point-process methods in conditioned aggregation models and may simplify calculations of absorption statistics or limiting densities under rotational symmetry.

major comments (2)

[§3.2] §3.2 (statement of the main Kurtz-type theorem): the intensity measure of the non-homogeneous PPP is asserted to be modulated by the survival probability up to the time-dependent radius r(t), but the manuscript does not explicitly verify that this modulation remains a deterministic function of position and time alone once the radius path is fixed; a short calculation showing that the absorption time for each particle is independent of all others given r(·) would strengthen the claim.
[§2.1] §2.1 (model definition): the pool is required to remain perfectly circular for all time by fiat, yet the absorption mechanism is driven by independent random walks whose hitting points are not uniformly distributed on the circle; the paper should confirm that the subsequent radius update (area proportional to mass) preserves the circular shape almost surely, as any deviation would invalidate the rotational symmetry used throughout the argument.

minor comments (2)

[Section 2] The notation for the particle point measure and the radius process r(t) is introduced only in the abstract and should be restated with a short table of symbols at the beginning of Section 2.
[Introduction] The reference list cites Kurtz (1978) but does not indicate which version of the theorem is being adapted; a one-sentence clarification in the introduction would help readers locate the precise statement used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address the two major comments point by point below.

read point-by-point responses

Referee: [§3.2] §3.2 (statement of the main Kurtz-type theorem): the intensity measure of the non-homogeneous PPP is asserted to be modulated by the survival probability up to the time-dependent radius r(t), but the manuscript does not explicitly verify that this modulation remains a deterministic function of position and time alone once the radius path is fixed; a short calculation showing that the absorption time for each particle is independent of all others given r(·) would strengthen the claim.

Authors: We thank the referee for highlighting this point. In the Pool model, once the radius path r(·) is fixed (deterministic, as it depends only on the cumulative mass via the area-proportional rule), each particle evolves as an independent continuous-time random walk. The absorption time τ for a particle starting at x is the first hitting time to the moving boundary |B_s| = r(s). Because the boundary path is the same for all particles and the walks are independent, the τ_i are conditionally independent given r(·). The survival probability up to time t, i.e., P(τ > t | x, r(·)), is then a deterministic functional of position x and time t (computed from the fixed path r(·)). We will add a short clarifying paragraph in the revised §3.2 explicitly stating this conditional independence and the resulting deterministic intensity. revision: yes
Referee: [§2.1] §2.1 (model definition): the pool is required to remain perfectly circular for all time by fiat, yet the absorption mechanism is driven by independent random walks whose hitting points are not uniformly distributed on the circle; the paper should confirm that the subsequent radius update (area proportional to mass) preserves the circular shape almost surely, as any deviation would invalidate the rotational symmetry used throughout the argument.

Authors: The Pool model is defined ab initio as a rotationally symmetric process: the pool is always the disk centered at the origin whose radius is updated at each absorption so that its area equals the current total mass. The shape is therefore circular by construction at every instant; the hitting locations of the random walks affect only the instants at which mass increases (and hence the radius growth schedule), but never the geometric shape itself. Consequently, no deviation from circularity can occur, and the rotational symmetry invoked in the analysis holds almost surely by the model definition in §2.1. We will insert one clarifying sentence in the revised §2.1 to make this explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from first principles

full rationale

The Pool model is introduced from first principles as a rotationally symmetric process in which independent continuous-time random walks are absorbed into a circular pool whose radius evolves deterministically with absorbed mass to preserve area scaling. The central result is presented as a version of Kurtz's theorem asserting that the particle configuration conditioned on a fixed pool-growth path is an independent non-homogeneous Poisson point process. This follows directly from the independence of particle trajectories together with the standard thinning property of Poisson point processes applied to the position-dependent survival probability induced by the deterministic radius function. No equation reduces to a fitted parameter renamed as a prediction, no self-citation is load-bearing for the uniqueness or the representation, and the rotational symmetry is an explicit modeling choice rather than a derived claim. The provided abstract and description contain no self-definitional steps or ansatzes smuggled via prior work by the same authors. The result is therefore independent of its inputs and does not reduce by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The model rests on standard assumptions for diffusion-limited aggregation and introduces the circular pool with mass-proportional expansion as its core invented element.

axioms (2)

domain assumption Particles perform continuous-time random walks in R^2
Standard modeling choice for diffusion-limited aggregation processes.
domain assumption Pool expands so its area grows proportionally to absorbed mass
Central definition ensuring mass preservation.

invented entities (1)

Pool model no independent evidence
purpose: Mass-preserving multi-particle aggregation process
Newly defined construction in the paper.

pith-pipeline@v0.9.0 · 5401 in / 1202 out tokens · 33845 ms · 2026-05-10T10:12:14.918934+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Allan Sly,On one-dimensional multi-particle diffusion limited aggregation, In and out of equi- librium 3: Celebrating vladas sidoravicius, 2020, pp. 755–774

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20, 11286–11295

Bairav S Vishnugopi, Feng Hao, Ankit Verma, and Partha P Mukherjee,Surface diffusion manifestation in electrodeposition of metal anodes, Physical Chemistry Chemical Physics22 (2020), no. 20, 11286–11295. 25 (Zhenhao Cai)F aculty of mathematics and computer science, Weizmann Institute of Science, Rehovot, Israel Email address:caizhenhao@pku.edu.cn (Eviatar...

work page 2020

[1] [1]

Berger, E

N. Berger, E. B. Procaccia, and D. Sharon,On one-dimensional cluster cluster model, arXiv preprint arXiv:2507.03552 (2025)

work page arXiv 2025

[2] [2]

Berger, E

N. Berger, E. B. Procaccia, and A. Turner,Growth of stationary hastings–levitov, The Annals of Applied Probability32(2022), no. 5, 3331–3360

work page 2022

[3] [3]

Noam Berger and Eviatar B Procaccia,Logarithmic fluctuations of stationary hastings-levitov, arXiv preprint arXiv:2502.03554 (2025)

work page arXiv 2025

[4] [4]

N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular variation, Encyclopedia of Mathe- matics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987

work page 1987

[5] [5]

3, 659–689

Lewis Bowen,Finitary random interlacements and the gaboriau–lyons problem, Geometric and Functional Analysis29(2019), no. 3, 659–689

work page 2019

[6] [6]

Zhenhao Cai, Yunfeng Xiong, and Yuan Zhang,On (non-) monotonicity and phase diagram of finitary random interlacement, Entropy23(2021), no. 1, 69

work page 2021

[7] [7]

Guanyi Chen, Eviatar B Procaccia, and Yuxuan Zong,One-arm domination time in cylindrical hastings-levitov(0), arXiv preprint arXiv:2507.11028 (2025)

work page arXiv 2025

[8] [8]

Richard Durrett,Probability : theory and examples / rick durrett, duke university., Fifth edi- tion., Cambridge series on statistical and probabilistic mathematics ; 49, Cambridge University Press, Cambridge, United Kingdom, 2019 2019 (eng)

work page 2019

[9] [9]

4, 682–686

Meyer Dwass,The total progeny in a branching process and a related random walk, Journal of Applied Probability6(1969), no. 4, 682–686

work page 1969

[10] [10]

5, 1838–1879

HARRY KESTEN and VLADAS SIDORAVICIUS,A problem in one-dimensional diffusion- limited aggregation (dla) and positive recurrence of markov chains, The Annals of Probability 36(2008), no. 5, 1838–1879

work page 2008

[11] [11]

3, Clarendon Press - Oxford, 1992

John Frank Charles Kingman,Poisson processes, Vol. 3, Clarendon Press - Oxford, 1992

work page 1992

[12] [12]

Meakin,Diffusion-limited aggregation in three dimensions: results from a new cluster-cluster aggregation model, Journal of colloid and interface science102(1984), no

P. Meakin,Diffusion-limited aggregation in three dimensions: results from a new cluster-cluster aggregation model, Journal of colloid and interface science102(1984), no. 2, 491–504

work page 1984

[13] [13]

,Multiparticle diffusion-limited aggregation with strip geometry, Physica A: Statistical Mechanics and its Applications153(1988), no. 1, 1–19

work page 1988

[14] [14]

Milan Paunovic and Mordechay Schlesinger,Fundamentals of electrochemical deposition, john wiley and sons, 2006

work page 2006

[15] [15]

Procaccia, Jiayan Ye, and Yuan Zhang,Percolation for the finitary random inter- lacements, ALEA18(2021), 265–287

Eviatar B. Procaccia, Jiayan Ye, and Yuan Zhang,Percolation for the finitary random inter- lacements, ALEA18(2021), 265–287

work page 2021

[16] [16]

H. B. Rosenstock and C. L. Marquardt,Cluster formation in two-dimensional random walks: Application to photolysis of silver halides, Physical Review B22(1980), no. 12, 5797

work page 1980

[17] [17]

2, 491–571

Vladas Sidoravicius and Alexandre Stauffer,Multi-particle diffusion limited aggregation, In- ventiones mathematicae218(2019), no. 2, 491–571

work page 2019

[18] [18]

Allan Sly,On one-dimensional multi-particle diffusion limited aggregation, In and out of equi- librium 3: Celebrating vladas sidoravicius, 2020, pp. 755–774

work page 2020

[19] [19]

20, 11286–11295

Bairav S Vishnugopi, Feng Hao, Ankit Verma, and Partha P Mukherjee,Surface diffusion manifestation in electrodeposition of metal anodes, Physical Chemistry Chemical Physics22 (2020), no. 20, 11286–11295. 25 (Zhenhao Cai)F aculty of mathematics and computer science, Weizmann Institute of Science, Rehovot, Israel Email address:caizhenhao@pku.edu.cn (Eviatar...

work page 2020