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.

Cylindrical Hastings Levitov

“…Note that in [1] the slit map on H is defined as √ z 2 − δ 2 . Here we need i to be a fixed point for ϕ δ (•), otherwise infinity on the cylinder would not be mapped to infinity on the cylinder after adding the slit.…”

“…Recently, Berger, Procaccia, and Turner [1,8] constructed a stationary version of the Hastings-Levitov model (SHL(0)) defined on the upper half-plane. They showed that unlike the Hastings Levitov process grown on a disk [7], particle sizes are tight without normalization, and thus SHL(0) is a good candidate for an off-lattice version of stationary DLA [9,10,11,12].…”

“…This is an off-lattice version of the result in [6], where it is shown that the classical discrete DLA initiated in a long line segment or a discrete cylinder converges to the Stationary DLA process as the length of the initial line tends to infinity. This allows to use the exact growth bounds proved in [1] for the more physical spatially finite setting of SHL on a cylinder. Moreover, the main result of this paper also shows that HL(0) in the small particle limit δ → 0, grown up to time tδ −1 , scaled around a point in H such that particles are of size 1 (i.e.…”

“…One can define CHL N (α), by normalizing the slit sizes according to the conformal radius. In [1, Question 1] it is conjectured that SHL(α) is not well defined for α ∈ [1,2]. Though not a full proof, it would be indicative to show that CHL N (α) has no natural limiting process as N → ∞, for any α ∈ [1,2].…”

“…Does vertical stationarity hold for the CHL N ? For self containment and uniformity of definitions we recall the definition of the SHL(0) appearing in [1].…”