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Tip growth in a strongly concentrated aggregation model follows local geodesics

Frankie Higgs

TL;DR

The paper rigorously connects two planar growth models, showing that the small-particle limit of the Aggregate Loewner Evolution (ALE) converges to the Laplacian path model (LPM). By analyzing Loewner's equation near singularities and extending martingale techniques, it handles tip singularities and constructs a chain of approximations—ALE to an auxiliary tip-attachment model, to a multinomial model, and finally to the LPM—proving convergence of driving measures in the bounded Wasserstein sense and of clusters in the Carathéodory topology. The approach blends conformal mapping techniques, backward Loewner dynamics, and singular Grönwall-type estimates to control errors across multiple scales, yielding a rigorous geodesic-growth limit for tip propagation. This establishes, under precise scaling of parameters and initial conditions, that tip growth in strongly regularised ALE becomes geodesic growth along hyperbolic geodesics toward infinity, as captured by the LPM. The results advance understanding of screening effects, harmonic-measure–driven selection among multiple arms, and the relationship between discrete-attachment models and continuum growth processes with conformal-invariant structure.

Abstract

We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of $k$ needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov in 2002, in which the tips grow along geodesics towards $\infty$. The proof of this result involves analysis of Loewner's equation near its singular points, and we extend usual martingale methods to this setting. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner's equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.

Tip growth in a strongly concentrated aggregation model follows local geodesics

TL;DR

The paper rigorously connects two planar growth models, showing that the small-particle limit of the Aggregate Loewner Evolution (ALE) converges to the Laplacian path model (LPM). By analyzing Loewner's equation near singularities and extending martingale techniques, it handles tip singularities and constructs a chain of approximations—ALE to an auxiliary tip-attachment model, to a multinomial model, and finally to the LPM—proving convergence of driving measures in the bounded Wasserstein sense and of clusters in the Carathéodory topology. The approach blends conformal mapping techniques, backward Loewner dynamics, and singular Grönwall-type estimates to control errors across multiple scales, yielding a rigorous geodesic-growth limit for tip propagation. This establishes, under precise scaling of parameters and initial conditions, that tip growth in strongly regularised ALE becomes geodesic growth along hyperbolic geodesics toward infinity, as captured by the LPM. The results advance understanding of screening effects, harmonic-measure–driven selection among multiple arms, and the relationship between discrete-attachment models and continuum growth processes with conformal-invariant structure.

Abstract

We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov in 2002, in which the tips grow along geodesics towards . The proof of this result involves analysis of Loewner's equation near its singular points, and we extend usual martingale methods to this setting. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner's equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.
Paper Structure (14 sections, 27 theorems, 223 equations, 2 figures)

This paper contains 14 sections, 27 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Loewner's equation
  3. Particle aggregation models
  4. Aggregate Loewner evolution
  5. Laplacian path model
  6. Main result
  7. Analytic tools
  8. Removing the regularisation
  9. ALE to auxiliary model
  10. Auxiliary to multinomial model
  11. Multinomial to Laplacian path model
  12. Lipschitz bound on the locations and probabilities
  13. Grönwall-type bound on the total error
  14. Proof of main theorem

Key Result

Theorem 1.1

For a fixed $T > 0$, let $(\Phi^{\mathrm{ALE}}_t)_{t\in[0,T]}$ be the ALE($\alpha,\eta, \sigma$) map started from the initial cluster $K_0 \cup \overline{\mathbb{D}}$, where $K_0 = \bigcup_{j=1}^{k} e^{i\phi^j_0}(1, 1+d_j]$ for $d_j > 0$ and distinct $\phi^j_0 \in [0,2\pi)$. Let $\mu^{\mathrm{ALE}}_

Figures (2)

  • Figure 1: Simulations of the aggregate Loewner evolution started from a non-trivial configuration of arms. The initial cluster is drawn with thin black lines, and the attached particles are coloured by arrival time (the lighter particles arrived earlier). In the top-left diagram $\sigma$ is large enough that some particles are not attached at the tips of the existing slits and we see "branching" behaviour which does not occur in the limiting regime. In the three diagrams on the left one slit seems to grow much slower than the rest, agreeing with the observation in lpm that there is competition between the arms in the Laplacian path model, with only a certain number (depending on $\eta$) likely to survive (i.e. growth to a length proportional to the diameter of the entire cluster) as $T \to \infty$. The code used for the simulations is available at https://github.com/frankiehiggs/ALE-from-slits.
  • Figure 2: One step in the construction of the auxiliary model. The large red dot represents the image of $\exp(i \theta_{6}^*)$ under each map. Comparing $K_5$ and $K_6$, we can see that both have three curves but the attachment points have been rotated from one cluster to the next. For this symmetric initial configuration and small capacity, the curves resemble slits, but this would not be true in general.

Theorems & Definitions (79)

  • Definition 1.1.1
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.2.1
  • Definition 1.3.1
  • Remark
  • Remark
  • Definition 1.4.1
  • Remark
  • Theorem 1.1
  • ...and 69 more