Fluctuation results for Hastings-Levitov planar growth

TL;DR

This paper analyzes the fluctuations of Hastings-Levitov clusters, showing they converge to a Gaussian process described by a stochastic fractional heat equation, revealing deep connections to log-correlated Gaussian fields.

Contribution

It provides an explicit construction of the Gaussian fluctuation process and characterizes its boundary behavior as an Ornstein-Uhlenbeck process driven by fractional heat dynamics.

Findings

Fluctuations are given by a continuous Gaussian process in holomorphic functions.

Boundary values follow an Ornstein-Uhlenbeck process on the circle.

Cluster boundary converges to a log-correlated fractional Gaussian field.

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 \[ \frac{\partial}{\partial t} \mathcal{W} (t,\vartheta ) = - (-\Delta )^{1/2} \mathcal{W} (t,\vartheta ) + \sqrt{2}\, \xi (t, \vartheta ) \,, \] 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 $\mathbb{T}$.