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arxiv: 2604.21142 · v1 · submitted 2026-04-22 · 🧮 math.PR

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Gaussian fluctuations for Internal DLA on cylinders

Ahmed Bou-Rabee , Vittoria Silvestri , Ariel Yadin
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:03 UTC · model grok-4.3

classification 🧮 math.PR
keywords internal DLAGaussian free fieldcylinder graphsfluctuationsshape theoremvertex-transitive graphseigenvalue convergencerandom growth
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The pith

Average fluctuations of internal DLA on cylinders converge to the Gaussian free field whenever the base graph is vertex-transitive and satisfies eigenvalue convergence.

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

The paper shows that internal DLA on the cylinder V_N times the line has average fluctuations matching the Gaussian free field for any vertex-transitive base graph V_N that meets a spectral convergence condition. It also derives a stronger bound on the largest deviations of the cluster from its mean position. This bound is enough to prove that the overall shape of the cluster settles to a fixed limit. A reader would care because the Gaussian character of the fluctuations now appears for many more base graphs than just the cycle, pointing to a mechanism that depends mainly on symmetry and spectral behavior rather than the exact lattice structure.

Core claim

For any vertex-transitive base graph V_N satisfying an eigenvalue convergence condition, the averaged fluctuations of the internal DLA cluster on the cylinder V_N × ℤ are described by the Gaussian free field. The paper also establishes an improved bound on the maximal fluctuations of the cluster, which is strong enough to yield a shape theorem asserting that the cluster converges to a deterministic limiting shape on every such cylinder.

What carries the argument

The eigenvalue convergence condition on the sequence of base graphs V_N, which transfers the harmonic analysis and Green function estimates from the cycle case to general vertex-transitive bases and thereby controls the IDLA height fluctuations.

If this is right

  • The IDLA cluster on V_N × ℤ converges to a deterministic limiting shape for every vertex-transitive base V_N.
  • The maximal height fluctuations of the cluster remain bounded by a term that vanishes after appropriate scaling.
  • The covariance structure of the averaged fluctuations matches that of the Gaussian free field in the scaling limit.
  • The shape theorem and fluctuation result hold uniformly for all bases meeting the stated symmetry and spectral conditions.

Where Pith is reading between the lines

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

  • The same spectral transfer technique could extend to IDLA on other product geometries or to related growth models such as Eden clusters.
  • Numerical checks on non-lattice bases satisfying the eigenvalue condition would give direct evidence for the rate of convergence to the Gaussian free field.
  • The improved maximal-fluctuation bound may simplify proofs of shape theorems for IDLA in higher-dimensional or non-cylindrical settings.

Load-bearing premise

The base graph V_N must be vertex-transitive and its eigenvalues must converge appropriately as the size of V_N grows.

What would settle it

A concrete sequence of vertex-transitive graphs satisfying the eigenvalue convergence condition for which the scaled average IDLA fluctuations on the cylinder fail to converge in law to the Gaussian free field.

Figures

Figures reproduced from arXiv: 2604.21142 by Ahmed Bou-Rabee, Ariel Yadin, Vittoria Silvestri.

Figure 1
Figure 1. Figure 1: Three samples of the IDLA cluster A(T) on the cylinder GN = VN × Z for different vertex-transitive base graphs VN : the 15 × 15 square lattice torus (N = 225), the 9×9 triangular lattice torus (N = 81), and the Nauru graph GP(12, 5) (N = 24). In each image, every site of the cluster is drawn as a prism whose cross-section is the Voronoi cell of its base vertex in the depicted embedding. converge to the GFF… view at source ↗
read the original abstract

Internal DLA is a discrete random growth model describing growing clusters of particles. Its limiting shape and fluctuations are well understood when the underlying graph is the $d$-dimensional lattice or the cylinder $\mathbb{Z}_N \times \mathbb{Z}$. In the latter geometry, the average fluctuations of IDLA have been shown to converge to the GFF. In this note we generalise this result by showing that, for any vertex-transitive base graph $V_N$ satisfying an eigenvalue convergence condition, the average fluctuations of IDLA on the cylinder $V_N \times \mathbb{Z}$ are given by a GFF. On the way, we present an improved bound on the clusters' maximal fluctuations, which is of independent interest and which implies a shape theorem for IDLA on $V_N \times \mathbb{Z}$ for any vertex-transitive base graph $V_N$.

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

0 major / 3 minor

Summary. The paper generalizes results on Internal DLA fluctuations on cylinders. For vertex-transitive base graphs V_N satisfying an eigenvalue convergence condition, it shows that the average fluctuations of IDLA clusters on V_N × ℤ converge to a Gaussian Free Field. It also establishes an improved bound on the maximal fluctuations of these clusters, which implies a shape theorem for IDLA on V_N × ℤ for arbitrary vertex-transitive V_N.

Significance. If correct, the results extend the GFF convergence for IDLA fluctuations from the cycle case (Z_N) to a natural broader class of vertex-transitive bases under a spectral assumption, while the improved maximal-fluctuation bound is of independent interest because it yields shape theorems without the eigenvalue condition. This strengthens the literature on random growth models by separating the spectral hypothesis (needed only for the GFF limit) from the shape result.

minor comments (3)
  1. The abstract and introduction should briefly compare the new maximal-fluctuation bound to the best previously known bounds (e.g., those for Z_N × ℤ) so that the improvement is quantified rather than asserted.
  2. Notation for the eigenvalue convergence condition on V_N should be introduced with a displayed definition or equation in the introduction, together with a short remark on why vertex-transitivity is used.
  3. The manuscript would benefit from an explicit statement of the error term or rate in the GFF approximation that follows from the eigenvalue condition, even if only in the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No specific major comments were provided in the report, so we have nothing to address point by point. We will incorporate any minor editorial suggestions from the editor or referee in the revised version.

Circularity Check

0 steps flagged

No significant circularity; result extends prior literature via external assumptions

full rationale

The derivation generalizes the known GFF limit for IDLA on Z_N x Z to vertex-transitive V_N with an eigenvalue convergence condition. This condition is stated as an external hypothesis on the base graph, not derived internally. The improved maximal-fluctuation bound is presented separately and used only to obtain the shape theorem for all vertex-transitive V_N (without the spectral condition). No equation reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported solely via self-citation, and the central GFF statement does not rename or self-define its inputs. The paper is therefore self-contained against external benchmarks once the stated assumptions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information available from the abstract alone; the central claims rest on the eigenvalue convergence condition and standard properties of vertex-transitive graphs.

axioms (1)
  • domain assumption Eigenvalue convergence condition on the sequence of base graphs V_N
    Invoked to ensure the GFF limit holds in the generalized cylinder setting.

pith-pipeline@v0.9.0 · 5443 in / 1260 out tokens · 45874 ms · 2026-05-09T23:03:13.888380+00:00 · methodology

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Reference graph

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