Fluctuations and mixing for planar random growth
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We study two models of random growth by aggregation on the plane, which turn out to share similar asymptotic features.
The first part of this thesis focuses on the Hastings-Levitov model HL(0), according to which clusters of particles are built on the complex plane by iterated composition of random conformal maps. Following the scaling limit result of Norris and Turner (2012), who proved that the limiting shape of large HL(0) clusters is a disc, we show that the fluctuations around this deterministic shape are described by a random holomorphic Gaussian field
The same scaling limit and boundary fluctuations are found by Jerison, Levine and Sheffield (2014) to arise in a different growth model, namely Internal Diffusion Limited Aggregation (IDLA). According to this discrete model, the aggregation process defines a Markov Chain on the infinite space of IDLA configurations, for which Jerison, Levine and Sheffield ask the following mixing question: how long does it take for IDLA dynamics to essentially forget where it started? We provide a partial answer to this question in the second part of this thesis, using coupling techniques to obtain an upper bound for this forget time. Finally, we specialise to IDLA on the cylinder graph