Random walks, diffusion limited aggregation in a wedge, and average conformal maps

TL;DR

This paper explores diffusion-limited aggregation in wedge geometries and examines how ensemble averages of conformal maps relate to noise-free shapes like the Saffman-Taylor finger, revealing complexities in noisy growth patterns.

Contribution

It demonstrates that ensemble averages of conformal maps in wedge geometries produce shapes close to, but not identical to, the Saffman-Taylor finger, challenging previous assumptions.

Findings

Average conformal maps yield shapes near the Saffman-Taylor finger.

DLA in wedge geometry differs from channel geometry results.

No simple relation exists between noisy DLA averages and noise-free solutions.

Abstract

We investigate diffusion-limited aggregation (DLA) in a wedge geometry. Arneodo and collaborators have suggested that the ensemble average of DLA cluster density should be close to the noise-free selected Saffman-Taylor finger. We show that a different, but related, ensemble average, that of the conformal maps associated with random clusters, yields a non-trivial shape which is also not far from the Saffman-Taylor finger. However, we have previously demonstrated that the same average of DLA in a channel geometry is not the Saffman-Taylor finger. This casts doubt on the idea that the average of noisy diffusion-limited growth is governed by a simple transcription of noise-free results.