The ALE (Aggregate Loewner Evolution) models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off-lattice analogues of several fundamental discrete models, such as Eden growth, Dielectric Breakdown and Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In these talks I will introduce the ALE models, which include Hastings-Levitov models as particular cases, and discuss their large scale properties such as scaling limits and fluctuations. I will conclude by stating some conjectures and open questions. Based on joint work with James Norris (Cambridge, UK) and Amanda Turner (Leeds, UK).
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