Discrete Laplacian Growth: Linear Stability vs Fractal Formation

TL;DR

This paper explores the relationship between stability and fractal formation in Discrete Laplacian Growth, revealing that fractal patterns are unaffected by the stability of the deterministic model, enabling a qualitative analysis.

Contribution

It introduces stochastic Discrete Laplacian Growth and analyzes its connection to the deterministic free-boundary problem, highlighting the independence of fractal growth from stability.

Findings

Fractal growth is unaffected by the stability of the deterministic model.

A qualitative analytic description of Discrete Laplacian Growth is provided.

Discrete growth exhibits fractal patterns regardless of stability conditions.

Abstract

We introduce stochastic Discrete Laplacian Growth and consider its deterministic continuous version. These are reminiscent respectively to well-known Diffusion Limited Aggregation and Hele-Shaw free boundary problem for the interface propagation. We study correlation between stability of deterministic free-boundary problem and macroscopic fractal growth in the corresponding discrete problem. It turns out that fractal growth in the discrete problem is not influenced by stability of its deterministic version. Using this fact one can easily provide a qualitative analytic description of the Discrete Laplacian Growth.