Vittoria Silvestris: Hastings-Levitov growth and the GFF

Abstract: The so called Hastings-Levitov (HL) planar aggregation models consist of growing random clusters in the complex plane, built by iterated composition of random conformal maps. When these maps are chosen in i.i.d. fashion, it was proved by Norris and Turner that the limiting shape of HL clusters is a disc. In this talk I will consider fluctuations around this asymptotic behaviour, showing that these are given by a random holomorphic Gaussian field F, which can be explicitly constructed. The boundary values of F perform a Gaussian Markov process in the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of a (complex) Gaussian Free Field to the unit circle.