Open Access
2023 Scaling limits of anisotropic growth on logarithmic time-scales
George Liddle, Amanda Turner
Author Affiliations +
Electron. J. Probab. 28: 1-32 (2023). DOI: 10.1214/23-EJP964


We study the anisotropic version of the Hastings-Levitov model AHL(ν). Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.

Funding Statement

Supported by EPSRC grant EP/T027940/1.


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George Liddle. Amanda Turner. "Scaling limits of anisotropic growth on logarithmic time-scales." Electron. J. Probab. 28 1 - 32, 2023.


Received: 8 November 2022; Accepted: 22 May 2023; Published: 2023
First available in Project Euclid: 2 June 2023

arXiv: 2211.03676
MathSciNet: MR4596350
zbMATH: 1519.60042
Digital Object Identifier: 10.1214/23-EJP964

Primary: 60Fxx , 60K35

Keywords: anisotropic growth , Fluctuations , Planar random growth , scaling limits

Vol.28 • 2023
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