Theory of Diffusion Controlled Growth

TL;DR

This paper develops a comprehensive theoretical framework for diffusion-controlled growth models, connecting them with turbulence analogies and predicting new phenomena like a Screening Transition, supported by analytical calculations and prior simulation data.

Contribution

It introduces a unified theoretical approach to diffusion-limited aggregation and dielectric breakdown models, including mode coupling and superuniversality, with novel predictions such as the Screening Transition.

Findings

Analytical calculations align with previous simulation results.

Identification of superuniversality in tip scaling exponents.

Prediction of a new Screening Transition at large eta.

Abstract

We expand upon a new theoretical framework for Diffusion Limited Aggregation and associated Dielectric Breakdown Models in two dimensions [R. C. Ball and E. Somfai, Phys. Rev. Lett. 89, 135503 (2002)]. Key steps are understanding how these models interrelate when the ultra-violet cut-off strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new Screening Transition in the DBM at large eta.