Conformal theory of the dimensions of diffusion-limited aggregates

We employ the recently introduced conformal iterative construction of Diffusion-Limited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process which is complementary to the iterative cluster growth. We use this method to establish the existence of a series of random scaling functions that yield, via the thermodynamic formalism of multifractals, the generalized dimensions D q of DLA for q ⩾ 1. The scaling function is determined just by the last stages of the iterative growth process which are relevant to the complementary dynamics. Using the scaling relation D 3 = D 0 /2, we estimate the fractal dimension of DLA to be D 0 = 1.69 ± 0.03.