Stability of regularized Hastings-Levitov aggregation in the subcritical regime

Abstract

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE$(\alpha,\eta)$ of continuum planar aggregation models. The class includes regularized versions of the Hastings--Levitov family HL$(\alpha)$ and continuum versions of the family of dielectric breakdown models, where the local attachment intensity for new particles is specified as a negative power $-\eta$ 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 $\zeta=\alpha+\eta$. Our results are subject to a subcriticality condition $\zeta\le1$: this includes HL$(\alpha)$ for $\alpha\le1$ and also the case $\alpha=2,\eta=-1$ corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL$(\alpha)$ at $\alpha=1$, 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 $\zeta\le1$.