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

We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of ROSENSTOCK and MARQUARDT on the plane. Based on the findings of (Ann. Probab. 24 (1996) 559–598, Arch. Ration. Mech. Anal. 233 (2019) 643–699, Delarue, Nadtochiy and Shkolnikov (2019)) 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 for the heat equation. To address this conjecture, we first prove that the limit points of diffusively scaled MDLA systems are well-defined and described by absorbed Brownian motions. Then, we show that these limit points satisfy the equation that characterizes the growth rate of the solid phase in the supercooled Stefan problem with an inequality, which can be strict in general. This result provides the first rigorous answer to a question that has received much attention in the physics literature. In the course of the proof, we establish two additional results interesting in their own right: (i) the stability of a “crossing property” of planar Brownian motion and (ii) a rigorous connection between the probabilistic solutions to the supercooled Stefan problem and its classical and weak solutions.