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 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 ∂/∂W(t,ϑ)=-(-Δ)1/2W(t,ϑ)+√2ξ(t,ϑ),where Δ denotes the Laplace operator acting on the spatial component, and ξ(t, ϑ) 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 (-Δ) -1/4W, for W complex white noise on T.