Fractal Structure of Hastings-Levitov Patterns Restricted in a Sector Geometry

TL;DR

This paper explores how restricting the Hastings-Levitov algorithm to a sector geometry influences the fractal patterns formed, revealing dependencies on sector angle and measure distribution, and comparing properties with other DLA models.

Contribution

It introduces a generalized HL algorithm for sector-restricted DLA, analyzing fractal properties and anisotropy, and compares different measure distributions' effects on pattern morphology.

Findings

Fractal dimension depends on sector angle $eta$ for uniform measure.

Sinusoidal measure yields fractal dimension similar to radial DLA.

Anisotropy and pattern appearance align with ADLA clusters.

Abstract

A generalized form of the Hastings and Levitov (HL) algorithm for simulation of diffusion-limited aggregation (DLA) restricted in a sector geometry is studied. It is found that this generalization with uniform measure produces "wedge-like" fractal patterns in the physical space, whose fractal dimension and anisotropy exponent depend significantly on the opening angle $\beta$ of the sector. The morphological properties and the overall shape of the patterns are analyzed by computing the angular two-point density correlation function of the patterns. We also find that the fractal dimension of the patterns with sinusoidal distributed measure depend weakly on $\beta$ with almost the same dimension as the radial DLA cluster. The anisotropy exponent and the visual appearance of the patterns in this case are shown to be compatible with those of the advection-diffusion-limited aggregation (ADLA) clusters.