Exact solution of diffusion limited aggregation in a narrow cylindrical geometry

TL;DR

This paper provides an exact analytical solution for diffusion limited aggregation (DLA) and dielectric breakdown models (DBM) in a narrow cylindrical geometry, accurately estimating fractal dimensions and confirming previous approximate methods.

Contribution

It introduces an exact solution method for DLA and DBM in cylindrical geometry, advancing understanding of their fractal properties.

Findings

Exact steady-state solutions for DLA and DBM in cylindrical geometry.

Fractal dimension estimates closely match simulation results.

Results agree with previous approximate fixed scale transformation methods.

Abstract

The diffusion limited aggregation model (DLA) and the more general dielectric breakdown model (DBM) are solved exactly in a two dimensional cylindrical geometry with periodic boundary conditions of width 2. Our approach follows the exact evolution of the growing interface, using the evolution matrix E, which is a temporal transfer matrix. The eigenvector of this matrix with an eigenvalue of one represents the system's steady state. This yields an estimate of the fractal dimension for DLA, which is in good agreement with simulations. The same technique is used to calculate the fractal dimension for various values of eta in the more general DBM model. Our exact results are very close to the approximate results found by the fixed scale transformation approach.