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Fractal “Aggregates” in the Complex Plane

Aggregates generated by probabilistic diffusion are fractals reminiscent of the deterministically generated Julia sets: the growth measure of the aggregate is equivalent to the “electric field” around the Julia set and both fractals have a characteristic hairy structure with the fields diverging at the tips. We conjecture that their f(α) spectra share the same qualitative features. For Julia sets the part of the spectrum corresponding to the tips (smallest α's) is robust under small parameter changes. In the fjords the electric field is vanishingly small, the associated α's diverge and f(α)'s are noisy, poorly convergent and highly sensitive to small variations of parameters. The fjords are screened, and this screening can manifest itself in f(α) as a “phase transition” at the Hausdorff dimension.