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

TL;DR

This paper investigates the scaling limits of a multi-particle diffusion-limited aggregation model on the plane, proposing it converges to a supercooled Stefan problem's solid phase, extending previous one-dimensional results to multiple dimensions.

Contribution

It extends the probabilistic formulation of the supercooled Stefan problem to multiple dimensions and links the MDLA process to this PDE-based growth model.

Findings

The scaling limit of the MDLA process satisfies the Stefan problem growth equation with an inequality.

Established stability of a crossing property of planar Brownian motion.

Connected probabilistic solutions to classical and weak solutions of the Stefan problem.

Abstract

We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of Rosenstock and Marquardt on the plane. Based on the recent findings of [11], [10] 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 extend the probabilistic formulation from [10] to multiple space dimensions. We then show that the equation that characterizes the growth rate of the solid phase in the supercooled Stefan problem is satisfied by the scaling limit of the external MDLA process with an inequality, which can be strict in general. 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.