One-Dimensional Scaling Limits in a Planar Laplacian Random Growth Model

Abstract

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by |Φ ′ n | −η , where Φ n is the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for η > 1, aggregating particles attach to their immediate predecessors with high probability, while for η < 1 almost surely this does not happen. * sola@math.su.se † a.g.turner@lancaster.ac.uk

SLE scaling limits for a Laplacian random growth model

“…Asymptotically β(c) ∼ d(c) ∼ 2c 1/2 as c → 0. These explicit expressions are found in [15] and [20].…”

“…The aggregate Loewner evolution model introduced in [20] is a conformal aggregation model as in Section 1.1, where we choose the angle sequence (θ n ) n∈N such that the attachment angle θ n+1 of the (n + 1)th particle is a random variable whose distribution, conditional on (θ 1 , • • • , θ n ), depends on (an approximation of) the density of harmonic measure on the boundary of the nth cluster K n , and the nth particle we attach has a capacity c n+1 which is a function of the density of harmonic measure at the attachment point on ∂K n . The conditional distribution of θ n+1 and the way we obtain c n+1 are respectively controlled by the two parameters η and α.…”

“…We use models of conformal growth to overcome this problem by working in a space without any underlying anistropy, in this case the complex plane C. We typically start with an isotropic seed, such as the disc, and then add particles at positions chosen according to a probability distribution without any natural anisotropy. In this paper we will study the aggregate Loewner evolution (ALE(α, η)) model introduced in [20], which is a generalisation of the Hastings-Levitov process (HL(α)) [7].…”

Coexistence in a random growth model with competition

“…The HL(0) process mentioned above is obtained by taking (θ n ) n∈N to be i.i.d uniform [−1, 1) random variables, c n = c and t n = n for all n ∈ N. A continuous-time embedding of HL(0) is obtained by taking (θ n ) n∈N and (c n ) n∈N as before, but letting (t n ) n∈N 0 be a constant-rate Poisson process. For other choices in the literature, see [9] and the references therein.…”

“…SinceX c t ∼ π c for all t, X c t ∼ π c for all t T and therefore lim t→∞ X c t ∼ π c as required. It is therefore sufficient to construct a coupling in which T < ∞ a.s. By Corollary 2.9 in [5], (9) implies that for any coupling, the two processes X c t andX c t visit the compact set K at the same time infinitely often, almost surely. For each x ∈ K, let X c,x t denote the process with generator L c , starting from X c,x 0 = x.…”