Explosive growth for a constrained Hastings-Levitov aggregation model

TL;DR

This paper studies a constrained Hastings-Levitov aggregation model, revealing that the cluster grows explosively with diameter scaling as rac{ ext{sqrt}(t \, ext{log} n)}{}, showing a new instability in growth patterns.

Contribution

It proves that the constrained HL(0) model leads to explosive growth, with the cluster diameter scaling as rac{ ext{sqrt}(t \, ext{log} n)}{}, highlighting a novel instability phenomenon.

Findings

Cluster diameter scales as rac{ ext{sqrt}(t \, ext{log} n)}{}

Growth is highly concentrated around this scale

Explosive growth occurs once the cluster reaches positive capacity

Abstract

We consider a constrained version of the 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 $\sqrt{t\log n}$, uniformly in $t\in [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL$(0)$.