  The method of iterated conformal maps allows to study the harmonic measure of
Diffusion Limited Aggregates with unprecedented accuracy. We employ this method
to explore the multifractal properties of the measure, including the scaling of
the measure in the deepest fjords that were hitherto screened away from any
numerical probing. We resolve probabilities as small as $10^{-35}$, and present
an accurate determination of the generalized dimensions and the spectrum of
singularities. We show that the generalized dimensions $D_q$ are infinite for
$q<q^*$, where $q^*$ is of the order of -0.2. In the language of $f(\alpha)$
this means that $\alpha_{max}$ is finite. The $f(\alpha)$ curve loses
analyticity (the phenomenon of "phase transition") at $\alpha_{max}$ and a
finite value of $f(\alpha_{max})$. We consider the geometric structure of the
regions that support the lowest parts of the harmonic measure, and thus offer
an explanation for the phase transition, rationalizing the value of $q^*$ and
$f(\alpha_{max})$. We thus offer a satisfactory physical picture of the scaling
properties of this multifractal measure.
