Thermodynamic Formalism of the Harmonic Measure of Diffusion Limited Aggregates: Phase Transition and Converged $f(\alpha)$

We study the nature of the phase transition in the multifractal formalism of the harmonic measure of Diffusion Limited Aggregates (DLA). Contrary to previous work that relied on random walk simulations or ad-hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as $10^{-70}$. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*= -0.17\pm 0.02$. In the language of $f(\alpha)$ this means that $\alpha_{max}$ is finite. We present a converged $f(\alpha)$ curve.