Gaussian fluctuations for Internal DLA on cylinders

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 $V_N$.

• The paper’s main contribution is establishing the Gaussian Free Field as the universal scaling limit for IDLA fluctuations on cylindrical graphs.
• It applies martingale central limit theorems and coupling arguments to achieve sharper maximal fluctuation bounds and prove spectral convergence.
• It generalizes lattice results to arbitrary vertex-transitive bases, providing novel insights into stochastic Laplacian growth models.

Gaussian Fluctuations for Internal DLA on Cylinders

Introduction and Motivation

The paper "Gaussian fluctuations for Internal DLA on cylinders" (2604.21142) investigates the fluctuation structure of the Internal Diffusion Limited Aggregation (IDLA) model in a class of product graphs with cylindrical geometry, specifically, on graphs of the form $V_N \times \mathbb{Z}$ where $V_N$ is a finite, connected, vertex-transitive graph with $N$ vertices. While the limiting shapes and fluctuation properties of IDLA have been previously analyzed on lattices and certain cylinder geometries (notably the base being the cycle), this work generalizes the scope to arbitrary vertex-transitive bases satisfying a spectral convergence criterion. The research aim is to identify and statistically characterize the average fluctuation fields, linking them to the Gaussian Free Field (GFF), and to provide improved maximal fluctuation bounds.

Model Description

The IDLA process on the cylinder $V_N \times \mathbb{Z}$ is initiated from a flat "half-cylinder" configuration $R_0$ (i.e., all points with non-positive vertical coordinate), and grows by aggregating one new site at each time $t$ by releasing a particle from the base layer $V_N \times \{0\}$ according to the uniform measure. The particle performs a lazy random walk on the cylinder graph, with equal probability for vertical and horizontal moves (the horizontal moves using the simple random walk kernel $P_N$ on $V_N$), until it leaves the existing aggregate and occupies a new site.

The base graph $V_N$ is assumed to be vertex-transitive, ensuring symmetry and homogeneity, with the lazy random walk transition kernel satisfying symmetry, laziness, and irreducibility.

Main Results

Maximal Fluctuations and Shape Theorem

The first significant result is a high-probability control on the maximal vertical fluctuations of the IDLA cluster $V_N$0 compared to its deterministic hydrodynamic profile, $V_N$1. The authors establish that, with probability tending to $V_N$2 as $V_N$3, the aggregate is sandwiched between layers $V_N$4, with the width

$V_N$5

where $V_N$6 is a "critical" time scale related to the mixing time of $V_N$7. Specifically, for all polynomially large time scales, the typical fluctuations are at most $V_N$8, a significant tightening over prior bounds for general base graphs. This bound is derived via martingale concentration inequalities (Freedman's inequality for inner fluctuations) and an outer bounding argument using coupling and the abelian property of IDLA.

The fluctuation result immediately yields a universal shape theorem: the aggregate fills the cylindrical shape up to height $V_N$9, up to a vanishing error in the large $N$0 limit, for all vertex-transitive bases.

Spectral Convergence and Gaussian Free Field Fluctuations

A core technical advance is the establishment that, under a spectral convergence condition on the base graph $N$1, the rescaled average fluctuations of the IDLA cluster converge, in a test-function sense, to the Gaussian Free Field (GFF) on a half-cylinder. The spectral requirement is that low-lying eigenvalues of the (appropriately rescaled) graph Laplacian $N$2 converge to those of the Laplacian operator on a limiting compact manifold or domain as $N$3, with $N$4 the relevant space rescaling.

Given a test function $N$5 supported on a finite number of base graph eigenmodes, the central limit theorem proved states that

$N$6

where $N$7 is the discrepancy field between the actual aggregate and its mean profile, and $N$8 is an explicit variance matching the Gaussian Free Field on the half-cylinder, including a memory effect from the flat initial interface. The proof relies on a martingale central limit theorem, using the discrete harmonic martingale structure of layerwise sums, with quantitative control on the replacement error for non-harmonic test functions via layerwise $N$9 bounds and a novel Efron–Stein argument (leveraging the abelian property and vertex-transitivity).

As $V_N \times \mathbb{Z}$0 (distance from the initial layer in the vertical direction), the slice variance converges to that of a fractional GFF of order $V_N \times \mathbb{Z}$1 on the base.

Generality and Field-Level Convergence

The main results hold for all vertex-transitive $V_N \times \mathbb{Z}$2 under the low-spectrum convergence condition, including discrete tori, cycles, and bases that admit smooth manifold limits. For tori of arbitrary dimension, the fluctuations converge modewise to those of the continuum GFF on a corresponding half-cylinder.

When uniform second-moment bounds are available (as for tori and cycles), the central limit theorem extends to field-level convergence in appropriate Sobolev spaces, realizing the limit as the GFF in the space of distributions on the base.

Implications and Theoretical Impact

Universality: The identification of the GFF as the universal scaling limit for average IDLA height fluctuations on cylinders with broad base graphs highlights a universality class dominated by algebraic and spectral, rather than geometric, features. This extends the reach of previous fluctuation results to a wide class of product geometries beyond the lattice case.

Improved Techniques: The approach avoids prior mixing/coupling limitations by working directly with uniformity from vertex-transitivity and martingale tools, enabling the sharper fluctuation control and more general geometric applicability.

Bridges to Continuum Models: The connection to continuum half-cylinder GFFs and the identification of fractional GFF limits underscores the convergence of discrete stochastic growth models and their macroscopic PDE descriptions relevant in the theory of Laplacian growth and associated free-boundary problems.

Proof Architecture

• Martingale CLT: The fluctuation limit is proved by showing that summing a discrete harmonic observable over the cluster yields a martingale, whose increments can be analyzed via the martingale central limit theorem. The necessary replacement of non-harmonic test functions is performed mode-by-mode, with error controlled using abelian invariance and automorphism structure.
• Maximal Bound: The maximal fluctuation bounds are established via a multi-layered argument: Freedman's inequality for the inner bound, alive/ghost decompositions for the outer bound, and a recursive coupling reduction for the regime of slow mixing.
• Spectral Identification: The spectral assumption is exploited to relate fluctuations in each mode to the appropriate Ornstein–Uhlenbeck processes for the Laplacian eigenfunctions, reproducing the covariance structure of the GFF.

Extensions and Future Directions

• Beyond Vertex-Transitive Bases: While the present work restricts to vertex-transitive graphs, the techniques may extend to more general bases with weaker symmetry, provided uniformity can be recovered via stochastic approximation or mixing arguments.
• Non-Homogeneous Geometries and Manifolds: There is theoretical interest in extending results to non-homogeneous discretizations, e.g., graph approximations of arbitrary Riemannian manifolds, or random planar maps, with potential links to Liouville quantum gravity and random geometry.
• Other Growth Models: The techniques illuminate the relationship between IDLA and other Laplacian growth models (divisible sandpile, rotor aggregation), suggesting avenues to study fluctuation limits and scaling behaviors in these related models using similar martingale and spectral tools.
• Critical Interfaces and Initial Data: Varying the initial configuration or the source measure remains an open area, with potential for non-Gaussian fluctuation behavior or crossover regimes (as indicated by results on branching IDLA and external DLA).

Conclusion

The paper rigorously establishes that Internal DLA aggregates on cylinder graphs with vertex-transitive bases exhibit fluctuations, in a broad sense, governed by the Gaussian Free Field. Both the size and the universality of average fluctuation behavior are quantified, and the connection to continuum probabilistic fields is made precise through spectral analysis. The findings not only improve upon earlier bounds but provide new methodologies that apply to a rich class of discrete growth models, opening the door for further developments in probabilistic geometric analysis and the study of stochastic Laplacian growth phenomena.