Growth of Stationary Hastings-Levitov

Abstract

We construct and study a stationary version of the Hastings-Levitov(0) model. We prove that, unlike in the classical HL(0) model, in the stationary case the size of particles attaching to the aggregate is tight, and therefore SHL( 0) is proposed as a potential candidate for a stationary off-lattice variant of Diffusion Limited Aggregation (DLA). The stationary setting, together with a geometric interpretation of the harmonic measure, yields new geometric results such as stabilization, finiteness of arms and arm size distribution. We show that, under appropriate scaling, arms in SHL(0) converge to the graph of Brownian motion which has fractal dimension 3/2. Moreover we show that trees with n particles reach a height of order n 2/3 , corresponding to a numerical prediction of Meakin from 1983 for the gyration radius of DLA growing on a long line segment.

Explosive growth for a constrained Hastings–Levitov aggregation model

“…The above model is a variant of the well-known Hastings-Levitov model HL(0), where the growth usually takes place outside the unit disc D={|z|≤1}, rather than on the upper half-plane H [4], [6], [14] and [15] (see however [3] by Berger, Procaccia and Turner, where this model is introduced in the upper half-plane).…”

Explosive growth for a constrained Hastings-Levitov aggregation model

“…We have so far focused for convenience and ease on the situation in the upper-half plane, but it is in fact more standard to consider HL(0) as an aggregation model outside the unit disc (the recent paper [3] by Berger, Procaccia and Turner being the exception). The model is defined similarly by replacing the upper-half plane H with the unit disc D, and the elementary maps describing the slits are obtained from F by conjugating with respect to a conformal map from C \ D to H. See e.g.…”

“…The above model is a variant of the well-known Hastings-Levitov model HL(0), where the growth usually takes place outside the unit disc D = {|z| ≤ 1}, rather than on the upper half-plane H [5,13,14] (see however [3] by Berger, Procaccia and Turner, where this model is introduced in the upper half-plane).…”

The dimension of Diffusion Limited Aggregates grown on a line

“…The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit and a well defined fractal dimension. Mathematical proofs of the main results are available in [1].…”

“…(ω). This process was proven to exist in [1], was denoted "stationary Hasting-Levitov(0)" and shown to define a conformal map. In addition, this process is invariant to horizontal shifts.…”