A Unified Computational Framework for Two Dimensional Diffusion Limited Aggregation via Finite-Size Scaling, Multifractality, and Morphological Analysis
Diffusion-Limited Aggregation (DLA), the canonical model for non-equilibrium fractal growth, emerges from the simple rule of irreversible attachment by random walkers. Despite four decades of study, a unified computational framework reconciling its stochastic algorithm, universal fractal dimension, multifractal growth measure, and finite-size effects remains essential for applications from materials science to geomorphology. Through large-scale simulations (clusters up to $N = 10^6$ particles) in two dimensions, we perform a tripartite analysis: (1) We establish a definitive finite-size scaling collapse, extracting the universal fractal dimension $D = 1.712 \pm 0.015$ and identifying the crossover to boundary-dominated growth at a scaled mass $x_0 \approx 0.10 \pm 0.02$. (2) We quantify the full multifractal spectrum of the harmonic measure ($\Delta\alpha \approx 1.13$), directly linking the stochastic algorithm to the deterministic Laplacian growth equation $\nabla^2 p = 0$ and explaining the screening effect via an exponential decay $\eta \sim e^{-r/\xi}$ with screening length $\xi = 22.7 \pm 0.8$ lattice units. (3) We provide a complete morphological characterization, revealing power-law branch length distributions ($\tau \approx 2.1$) and angular branching preferences ($\sim 72^\circ$). This work computationally validates DLA as a robust universality class and provides a scalable methodology for analyzing diffusion-controlled pattern formation across disciplines.