Internal DLA in Higher Dimensions

TL;DR

This paper demonstrates that in dimensions greater than two, the internal diffusion limited aggregation cluster is nearly spherical, with deviations bounded by a logarithmic factor, advancing understanding of its geometric properties.

Contribution

The work establishes a quantitative bound on the shape deviation of internal DLA clusters in higher dimensions, showing near-spherical growth with logarithmic error.

Findings

A(t) is approximately spherical in dimensions d > 2

Shape deviation is bounded by O(√log t)

Provides geometric insight into internal DLA growth in higher dimensions

Abstract

Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d > 2. We show that A(t) is approximately spherical, up to an O(\sqrt{\log t}) error.