In this talk we will discuss the recent progress on such random growth models as Diffusion Limited Aggregation (DLA) and Dielectric-Breakdown Model (DBM) in 2 and 3 dimensions. These models are believed to exhibit non-equilibrium growth, producing irregular fractal patterns. The main questions about these processes include finding their scaling limits and fractal dimensions. However, almost nothing is known rigorously.
The main result about these models is due to Kesten, who gave a non-trivial lower bound on the fractal dimension of DLA clusters. The main tool in his proof was the famous Beurling’s estimate.
We generalize this result to DBM and give a new proof of Kesten’s Theorem. Our proof does not rely on Beurling’s estimate. Instead, we exploit the connection between DBM growth properties and multifractal spectrum of the harmonic measure.
This talk is based on joint work with Stanislav Smirnov.