In 1998 the physicists Hastings and Levitov introduced a family of continuum models to describe a range of physical phenomena of planar aggregation/diffusion. These consist of growing random clusters on the complex plane, which are built by iterated composition of random conformal maps. It was shown by Norris and Turner (2012) that in the case of i.i.d. maps the limiting shape of these clusters is a disc: in this talk I will show that the fluctuations around this shape are given by a random holomorphic Gaussian field F on {|z| > 1}, of which I will provide an explicit construction. When the cluster is allowed to grow indefinitely, I will show that the boundary values of F converge to a distribution-valued Fractional Gaussian Field on the unit circle, which is log-correlated, and critical in a sense that I will explain.