The Limiting Shape for Drifted Internal Diffusion Limited Aggregation is a True Heat Ball

TL;DR

This paper demonstrates that drifted internal diffusion limited aggregation (iDLA) clusters converge to a shape called a true heat ball, which satisfies a mean value property for caloric functions, bridging probabilistic models and PDE theory.

Contribution

It establishes the existence and boundedness of the limiting shape for drifted iDLA, connecting stochastic growth models with PDE concepts like heat balls.

Findings

Normalized cluster converges to a true heat ball shape.

True heat ball satisfies a mean value property for caloric functions.

Answers an open question in PDE theory about shape existence and boundedness.

Abstract

We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S, which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers a natural yet open question in PDE theory.