Alan Sola: Scaling limits of anisotropic Hastings-Levitov clusters

The idea of using iterated random conformal maps to produce random planar sets with interesting properties appears in a number of contexts in mathematics and physics. A well-known example is $\textrm{HL}(\alpha)$, a one-parameter family of planar random growth models put forward by Hastings and Levitov in a 1998 Physica D paper.
This family, with $\alpha \in [0,2]$, is meant to function as an off-latice interpolation between the Eden model ($\alpha=1$) and Diffusion-limited aggregation ($\alpha=2$).

While computer simulations of $\textrm{HL}(\alpha)$ have been carried out and analyzed by a number of physicists, rigorous mathematical progress has been slower so far--one notable exception being a 2005 paper of S. Rohde and M. Zinsmeister. For instance, they obtain bounds on capacity growth for (regularized) clusters, and show that $\textrm{HL}(0)$ are one-dimensional in a certain sense. Very recently, Norris and Turner have discovered a surprising connection between scaling limits $\textrm{HL}(0)$ clusters and a stochastic object known as the Brownian web.

After a brief introduction to the general $\textrm{HL}(\alpha)$ model, I will present results obtained jointly with Fredrik Johansson Viklund
(Columbia) and Amanda Turner (Lancaster). We have studied an anisotropic version of $\textrm{HL}(0)$ that models growth where certain spatial directions are preferred. In a natural scaling limit, we establish convergence of clusters and boundary flows, associated with the inverse conformal maps to the circle, to deterministic sets and solutions to certain ODEs, respectively. We also characterize higher-order fluctuations around the limit ODE flow in terms of the anisotropy as solutions to certain SDEs. Tools used include fluid limit techniques from stochastic analysis and the Loewner equation from conformal mapping.