# On outer fluctuations for internal DLA

## TL;DR

This paper revises and clarifies the proof of outer fluctuations in internal DLA clusters, addressing a flaw in previous crossing probability estimates and providing a self-contained exposition of the corrected argument.

## Contribution

It corrects a flaw in the previous proof of outer fluctuations for internal DLA and offers a self-contained, clearer presentation of the argument.

## Key findings

- Corrected the proof of outer fluctuations in internal DLA
- Provided a self-contained exposition of the crossing probability estimate
- Clarified the role of fingering in the fluctuation analysis

## Abstract

We had established inner and outer fluctuation for the internal DLA cluster when all walks are launched from the origin. In obtaining the outer fluctuation, we had used a deep lemma of Jerison, Levine and Sheffield, which estimate roughly the possibility of fingering, and had provided a simple proof using an interesting estimate for crossing probability for a simple random walk. The application of the crossing probability to the fingering for the internal DLA cluster contains a flaw discovered recently, that we correct in this note. We take the opportunity to make a self-contained exposition.