Lower bounds on fluctuations for internal DLA

TL;DR

This paper establishes a lower bound on the fluctuations of the internal DLA model in dimensions two and higher, matching known upper bounds in dimensions three and above, and provides additional bounds in specific directions.

Contribution

It introduces a new lower bound for fluctuation errors in internal DLA, confirming the tightness of existing upper bounds in higher dimensions.

Findings

Lower bound for fluctuations of order sqrt(log n) in dimensions ≥ 2.

Matching upper bounds for fluctuations in dimensions ≥ 3.

Additional bounds for directional inner error fluctuations.

Abstract

We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the d-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. When n random walks are sent from the origin, we establish a lower bound for the inner and outer errors fluctuations of order square root of the logarithm of n. When dimension is larger or equal to three, this lower bound matches the upper bound recently obtained in independent works of \cite{AG2} and \cite{JLS2}. Also, we produce as a corollary of our proof of \cite{AG2}, an upper bound for the fluctuation of the inner error in a specified direction.