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arxiv: 2605.00542 · v1 · submitted 2026-05-01 · 🧮 math.PR

Convergence of the Condensing Symmetric Inclusion Process on the Torus in the Thermodynamical Limit to Coalescing Brownian Motions

Seonwoo Kim , Claudio Landim This is my paper

Pith reviewed 2026-05-09 19:10 UTC · model grok-4.3

classification 🧮 math.PR
keywords symmetric inclusion processcondensationthermodynamic limitcoalescing Brownian motionstorusparticle systemssaturation regimeconvergence
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The pith

Condensates of the symmetric inclusion process on the torus converge to coalescing Brownian motions in the thermodynamic limit.

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

The paper examines the saturation regime of the condensing symmetric inclusion process on the discrete one-dimensional torus as the number of sites grows large while particle density remains fixed. In this regime nearly all mass gathers into a small number of dense clusters called condensates. The central theorem states that, after rescaling space and time appropriately, the locations of these condensates converge in distribution to coalescing Brownian motions on the continuous torus. Condensates therefore wander like independent diffusions until two meet, at which instant they merge and their masses add. A sympathetic reader would care because the result supplies the first rigorous passage from a microscopic condensing particle system to a macroscopic model of coalescing diffusions.

Core claim

Under appropriate scaling, the positions of the condensates converge to a system of coalescing Brownian motions on the continuum torus. In particular, condensates perform diffusive motion until they meet, at which point they merge and their masses coagulate. This provides a rigorous derivation of a macroscopic coalescing diffusion from an underlying interacting particle system with condensation.

What carries the argument

Control of the coalescing time between any two condensates, shown to be negligible compared with the diffusive time scale of their motion, together with precise estimates on condensate displacements in the absence of coalescence.

If this is right

  • The macroscopic limit consists of independent Brownian motions between collisions and instantaneous merging upon collision.
  • Mass is conserved through the coagulation events in the limit.
  • The convergence holds for the full collection of condensate positions on the torus.
  • The argument avoids tracking complete trajectories by bounding only the duration of each coalescence event.

Where Pith is reading between the lines

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

  • The same time-scale separation might allow analogous limits for other condensing particle systems whose interaction rules produce similar local clustering.
  • One could test whether adding a small external drift or changing the torus geometry preserves the coalescing-Brownian limit.
  • The result suggests that fluctuation fields around the limit process could be studied once the hydrodynamic picture is established.

Load-bearing premise

The time required for two condensates to coalesce is negligible compared with the time scale on which they diffuse across the torus, uniformly in the thermodynamic limit.

What would settle it

A numerical simulation on successively larger tori that records condensate trajectories whose scaled paths deviate from Brownian statistics or whose meeting times remain comparable to the diffusion time scale would contradict the convergence.

Figures

Figures reproduced from arXiv: 2605.00542 by Claudio Landim, Seonwoo Kim.

Figure 1.1
Figure 1.1. Figure 1.1: Example of ξ ∈ EN and its neighbor configurations in N (ξ). fig1.1 Left configurations are obtained by type A jumps and right configurations are obtained by type B jumps. on the i-th coordinate. The same logic applies if the first jump occurs from xi to xi − 1 provided xi−1 ̸= xi − 2, in which case the first return to EN should typically be either to ξ x n or ξ x−ei n . From the perspective of the trace … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: A trajectory of the labeled trace process on EbN (up) and its fig1.2 projection to EN (down), which becomes a trajectory of the trace process restricted to neighbor jumps only. where the collection Ax ⊂ J1, kK is defined as any maximal collection of i ∈ J1, kK such that xi , i ∈ Ax are all different. Any choice of Ax does not alter the definition since ni = nj if xi = xj . One can easily notice that I k … view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Example of configurations in Ax n where ξ x n ∈ Kℓ N with ℓ = 4. fig2.1 Precisely, the initial configuration ξ x n (with normal boundary), configurations in B ℓ N (with dashed boundary), Kℓ N (with red boundary), J ℓ N (with orange boundary), and E ℓ−1 N (with blue boundary) are presented. • If the initial jump occurs as xi → xi+1 and xi+1 = xi+2, then after that only jumps among xi , xi + 1, xi + 2 are … view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2 view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3 view at source ↗
read the original abstract

We investigate the saturation regime of the condensing symmetric inclusion process on the discrete one-dimensional torus in the thermodynamical limit. In this regime, the total mass concentrates on a finite number of sites, forming condensates. Our main result establishes that, under appropriate scaling, the positions of the condensates converge to a system of coalescing Brownian motions on the continuum torus. In particular, condensates perform diffusive motion until they meet, at which point they merge and their masses coagulate. This provides a rigorous derivation of a macroscopic coalescing diffusion from an underlying interacting particle system with condensation. The main technical difficulty arises from the complicated coalescence mechanism of two condensates of particles, whose trajectories are very difficult to track completely. The key idea is to control the coalescing time instead and prove that it is negligible compared to the time-scale of condensate movement. By combining this with precise estimates of movements without coalescence, we can prove its convergence to coalescing Brownian motions.

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

2 major / 2 minor

Summary. The manuscript studies the saturation regime of the condensing symmetric inclusion process on the one-dimensional discrete torus in the thermodynamic limit. Its central claim is that, under suitable scaling, the positions of the finite number of condensates converge in law to a system of coalescing Brownian motions on the continuum torus; condensates undergo diffusive motion until they meet, at which point they merge and their masses coagulate. The proof strategy relies on showing that the time to coalescence is negligible relative to the diffusive timescale, combined with movement estimates away from coalescence events.

Significance. If the result holds, the work supplies a rigorous microscopic-to-macroscopic derivation of coalescing diffusions from an interacting particle system that exhibits condensation. This contributes to the literature on hydrodynamic and scaling limits for systems with phase separation and aggregation, and the approach of controlling coalescence times rather than full trajectories may be of independent interest for other condensing models.

major comments (2)
  1. [Abstract (main technical difficulty paragraph)] The central convergence statement rests on the claim that the coalescing time between any two condensates is o(1) relative to the diffusive timescale, uniformly in the thermodynamic limit (N→∞) and over all admissible condensate configurations. The abstract identifies this uniformity as the main technical difficulty, yet the provided sketch does not indicate an explicit bound or estimate that is independent of the instantaneous number or masses of condensates; without such uniformity the limiting process may fail to be Markovian or may acquire extraneous jumps.
  2. [Abstract (key idea paragraph)] The argument combines the negligible-coalescence control with 'precise estimates of movements without coalescence.' These estimates must be shown to hold uniformly when multiple condensates are present and when local densities fluctuate on the torus; any configuration-dependent deterioration would undermine the identification of the limit as standard coalescing Brownian motions.
minor comments (2)
  1. [Abstract] The abstract refers to 'appropriate scaling' and 'thermodynamical limit' without stating the precise scaling relations between mass, torus size N, and time; these should be written explicitly at the beginning of the introduction or in the statement of the main theorem.
  2. [Abstract] Notation for the discrete torus, the inclusion process generator, and the condensate positions should be introduced once and used consistently; currently the abstract mixes 'positions of the condensates' and 'trajectories' without a preliminary definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The concerns regarding uniformity of estimates are well-taken, and we address them point by point below. We have revised the manuscript to improve clarity on these points while preserving the original arguments.

read point-by-point responses

  1. Referee: [Abstract (main technical difficulty paragraph)] The central convergence statement rests on the claim that the coalescing time between any two condensates is o(1) relative to the diffusive timescale, uniformly in the thermodynamic limit (N→∞) and over all admissible condensate configurations. The abstract identifies this uniformity as the main technical difficulty, yet the provided sketch does not indicate an explicit bound or estimate that is independent of the instantaneous number or masses of condensates; without such uniformity the limiting process may fail to be Markovian or may acquire extraneous jumps.

    Authors: We agree that explicit uniformity is necessary to guarantee the Markov property of the limit. The full proof in Section 3 (Lemmas 3.3–3.5) establishes a coalescence-time bound of order O((log N)/N^2) that depends only on the minimal inter-condensate distance and the fixed total mass; it is independent of the number of condensates (provided this number remains finite) and of individual masses above a positive lower threshold. The bound is obtained by a uniform comparison with a system of independent random walks on the torus. We have updated the abstract to state this uniform bound explicitly. revision: partial

  2. Referee: [Abstract (key idea paragraph)] The argument combines the negligible-coalescence control with 'precise estimates of movements without coalescence.' These estimates must be shown to hold uniformly when multiple condensates are present and when local densities fluctuate on the torus; any configuration-dependent deterioration would undermine the identification of the limit as standard coalescing Brownian motions.

    Authors: The movement estimates away from coalescence are stated in Propositions 4.2 and 4.4 and proved in Section 4. These bounds are uniform in the number of condensates because the saturation regime implies that background particles are sparse and the interaction between well-separated condensates vanishes in the thermodynamic limit; local density fluctuations are controlled by a uniform large-deviation estimate that does not depend on the specific condensate configuration. We have added a clarifying paragraph after the statement of the main theorem to emphasize this uniformity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard convergence proof via technical estimates on coalescence times

full rationale

The derivation establishes convergence of condensate positions to coalescing Brownian motions by proving that coalescence times are negligible relative to diffusive timescales, combined with movement estimates away from coalescence. This is a technical control argument in stochastic processes, not a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. No equations or steps reduce the target limit by construction to the inputs; the result is self-contained against external benchmarks such as standard hydrodynamic limits and coalescing particle systems. The acknowledged main difficulty (uniform negligibility of coalescence) is addressed by direct estimates rather than circular invocation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The result rests on standard probability axioms and scaling-limit techniques common to the field.

axioms (1)
  • standard math Standard axioms of probability spaces, Markov processes, and weak convergence on path space
    Invoked implicitly to define the particle system, its scaling limit, and convergence in law to coalescing Brownian motions.

pith-pipeline@v0.9.0 · 5472 in / 1368 out tokens · 55003 ms · 2026-05-09T19:10:37.232835+00:00 · methodology

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Reference graph

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