Stability of regularized Hastings-Levitov aggregation in the subcritical regime

Understand

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.