Scaling limits of external multi-particle DLA on the plane and the supercooled Stefan problem

I will discuss a variant of the external multi-particle diffusion-limited aggregation (MDLA) process on the plane. Based on the recent findings in one space dimension it is natural to conjecture that the scaling limit of the growing aggregate in such a model is given by the growing solid phase in a suitable “probabilistic” formulation of the single-phase supercooled Stefan problem (1SSP). In the talk I will explain, on the one hand, how to extend the probabilistic formulation of the 1SSP to multiple space dimensions. On the other hand, I will argue that the scaling limits of external MDLA satisfy such a probabilistic formulation with an inequality, which can be strict in general. Time permitting I will discuss the “crossing property” of planar Brownian motion which forms the backbone of the proof. Based on joint work with Sergey Nadtochiy and Xiling Zhang.