Tip growth in a strongly concentrated conformal growth model follows local geodesics

Frankie Higgs

doi:10.1007/s00440-026-01494-w

Tip growth in a strongly concentrated conformal growth model follows local geodesics

Frankie Higgs

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

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 which the tips grow along geodesics towards
. Our proof involves analysis of Loewner’s equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. 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.

Original language	English
Number of pages	54
Journal	Probability Theory and Related Fields
Early online date	21 May 2026
DOIs	https://doi.org/10.1007/s00440-026-01494-w
Publication status	E-pub ahead of print - 21 May 2026

Data Availability Statement

No datasets were generated or analysed during the current study.

Acknowledgements

The initial version of this paper was completed while studying for a PhD at Lancaster University, and many thanks are due to my then-supervisor Amanda Turner for her guidance and patience. Thanks also to both examiners Dmitry Belyaev and Dmitry Korshunov who provided useful feedback. I benefited significantly from discussions with Alan Sola and Fredrik Viklund. An anonymous reviewer gave many suggestions which improved the presentation.

Funding

This work was begun under the Engineering and Physical Sciences Research Council studentship 2118765. Part of it was performed while the author was visiting the Mathematical Sciences Research Institute supported by the National Science Foundation (grant number DMS-1928930), and later while working under the EPSRC grant EP/T028653/1 and Royal Society grant RFERE231149.

Funders	Funder number
National Science Foundation	DMS-1928930
Engineering and Physical Sciences Research Council	EP/T028653/1
Royal Society	RFERE231149

Keywords

Conformal mapping
Harmonic measure
Random growth
Scaling limits
Martingales

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1007/s00440-026-01494-wLicence: CC BY

Cite this

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title = "Tip growth in a strongly concentrated conformal growth model follows local geodesics",

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 which the tips grow along geodesics towards . Our proof involves analysis of Loewner{\textquoteright}s equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner{\textquoteright}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.",

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KW - Conformal mapping

KW - Harmonic measure

KW - Random growth

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Tip growth in a strongly concentrated conformal growth model follows local geodesics

Abstract

Data Availability Statement

Acknowledgements

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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