Diffusion Limited Aggregation and Iterated Conformal Maps

Abstract

The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function $Φ^{(n)}$ which conformally maps the exterior of the unit circle to the exterior of an $n$-particle DLA. The map $Φ^{(n)}$ is obtained from $n$ stochastic iterations of a function $ϕ$ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of $Φ^{(n)}$, which asymptotically becomes a deterministic function of $n$. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.