Fluctuation results for Hastings–Levitov planar growth

Abstract

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 $$\mathcal {F}$$ taking values in the space of holomorphic functions on $$\{ |z|>1 \}$$ , of which we provide an explicit construction. The boundary values $$\mathcal {W}$$ of $$\mathcal {F}$$ are shown to perform an Ornstein–Uhlenbeck process on the space of distributions on the unit circle $$\mathbb {T}$$ , which can be described as the solution to the stochastic fractional heat equation $$\begin{aligned} \frac{\partial }{\partial t} \mathcal {W} (t,\vartheta ) = - (-\Delta )^{1/2} \mathcal {W} (t,\vartheta ) + \sqrt{2}\, \xi (t, \vartheta ), \end{aligned}$$ where $$\Delta $$ denotes the Laplace operator acting on the spatial component, and $$\xi (t,\vartheta )$$ is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process $$\mathcal {W}$$ converges to a log-correlated fractional Gaussian field, which can be realised as $$(-\Delta )^{-1/4}W$$ , for W complex white noise on $$\mathbb {T}$$ .