Arkiv för Matematik

Volume 61 (2023)

Number 1

Explosive growth for a constrained Hastings–Levitov aggregation model

Pages: 41 – 66



Nathanaël Berestycki (Department of Mathematics, University of Vienna, Austria)

Vittoria Silvestri (Department of Mathematics, University of Rome La Sapienza, Rome, Italy)


We consider a constrained version of the $\operatorname{HL}(0)$ Hastings–Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t \operatorname{log} n}$, uniformly in $t \in [0, T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained $\operatorname{HL}(0)$.

Received 23 November 2021

Received revised 7 October 2022

Accepted 20 October 2022

Published 26 April 2023