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 $\{\mathrm{\mid }z\mathrm{\mid }>1\}$, 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 $$\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathcal{W}(t,\vartheta )=-(-\mathrm{\Delta }{)}^{1\mathrm{/}2}\mathcal{W}(t,\vartheta )+\sqrt{2}\xi (t,\vartheta ),$$ where $\mathrm{\Delta }$ 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$W$complexWhiteNoiseon$$\mathbb{T}$$.