  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 $\mathbb{T}$.
