We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the attachment point of each successive particle is distributed according to harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability.
@cite_8: Abstract The Hastings–Levitov process HL ( α ) describes planar random compact subsets by means of random compositions of conformal maps. We prove the existence of the scaling limit of HL ( 0 ) and show that the limit sets are one-dimensional. We also give estimates for the dimension of HL ( α ) for 0 α ⩽ 2 , and discuss scaling limits of deterministic variants.
@cite_9: We study the fluctuations of the outer domain of Hastings–Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process ( F ) taking values in the space of holomorphic functions on ( |z|>1 ), of which we provide an explicit construction. The boundary values ( W ) of ( F ) are shown to perform an Ornstein–Uhlenbeck process on the space of distributions on the unit circle ( T ), which can be described as the solution to the stochastic fractional heat equation @math where ( ) denotes the Laplace operator acting on the spatial component, and ( (t, ) ) is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process ( W ) converges to a log-correlated fractional Gaussian field, which can be realised as ((- )^ -1 4 W ), for W complex white noise on ( T ).
@cite_1: We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.
@cite_2: We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.
@cite_13: We consider a family of growth models defined using conformal maps in which the local growth rate is determined by (| _n'|^ - ), where ( _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 ( >1 ), aggregating particles attach to their immediate predecessors with high probability, while for ( <1 ) almost surely this does not happen.
@cite_11: We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by c_n=c| n-1 '(e^ +i )|^ - , 0, where c>0 is the capacity of the first particle, _n are the composed conformal maps defining the clusters of the evolution, _n are independent uniform angles determining the positions at which particles are attached, and >0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c converges to 0, the clusters converge to growing disks with deterministic capacities, provided that does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting tend to 0 at different rates it converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.
In @cite_13 , Sola, Turner and Viklund showed the existence of a phase transition in the ALE( @math ) model at @math by focussing on the regime @math . They showed, in this case, that if particles are taken to be slits, and the regularisation parameter @math is sufficiently small, then the clusters converge to macroscopic growing slits. Almost all other previous work relates to HL(0). In @cite_1 , Norris and Turner showed that the HL(0) cluster converges to a disk with internal branching structure given by the Brownian web. More recently, Silvestri @cite_9 analysed the fluctuations in HL(0) and showed that these converge to a log-correlated fractional Gaussian field. Several other papers consider modifications of the HL(0) model @cite_2 @cite_11 @cite_8 .