  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.
