Stability of Regularized Hastings–Levitov Aggregation in the Subcritical Regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE ( α , η ) of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL ( α ) and continuum versions of the family of dielectric-breakdown models, where the l...

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Bibliographic Details
Published in:Communications in mathematical physics 2024-03, Vol.405 (3), Article 74
Main Authors: Norris, James, Silvestri, Vittoria, Turner, Amanda
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Dielectric breakdown
Mathematical and Computational Physics
Mathematical models
Mathematical Physics
Parameters
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Scaling
Theoretical
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Summary:We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE ( α , η ) of continuum planar aggregation models. The class includes regularized versions of the Hastings–Levitov family HL ( α ) and continuum versions of the family of dielectric-breakdown models, where the local attachment intensity for new particles is specified as a negative power - η of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner–Kufarev equation, where the driving measure is made to depend on the solution and on the parameter ζ = α + η . Our results are subject to a subcriticality condition ζ ⩽ 1 : this includes HL ( α ) for α ⩽ 1 and also the case α = 2 , η = - 1 corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL ( α ) at α = 1 , consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein–Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if ζ ⩽ 1 .
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-024-04960-5