Fluctuation results for Hastings–Levitov planar growth

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 \(\mathbb {T}\).