Simulations of External Multi-Particle DLA on the Plane and Connections to the Super-Cooled Stefan Problem

Abstract

We consider the external multi-particle difusion-limited aggregation (MDLA) process on the 2-dimensional integer grid. In this random growth process, particles are distributed uniformly at random on the grid and undergo a simple symmetric random walk with exclusion. They walk until they contact a collection of particles centered at the origin, at which point they attach. Iterating this process, a cluster forms. Since its inception, DLA models in the plane have resisted rigorous mathematical treatment. Whereas mathematicians have succeeded in establishing scaling limits for other stochastic growth models, this result for DLA process remains elusive. Recent findings in (1) establish a connection between the 1-dimensional MDLA process and solutions to a partial differential equation known as the super-cooled Stefan problem in space dimension (1SSP0. By fully characterizing the solutions to 1SSP, scaling limits for the 1-dimensional MDLA process were proven. It is natural to conjecture that a similar connection holds between 2-dimensional MDLA and SSP in two space dimensions. To address the conjecture, we take a numerical approach. We simulate the 2-dimensional MDLA process by decreasing the grid mesh towards 0. By studying the regularity of the cluster's interior and the statistical properties of its boundary, we show that the 2-dimensinoal MDLA process does not converge to solutions of 2SSP. We discuss why this process fails and propose possible resolutions.