RANDOM FRACTALS, PHASE-TRANSITIONS, AND NEGATIVE DIMENSION SPECTRA

We introduce an exactly solvable model of a random fractal which, for any finite resolution of the length scale l, exhibits a negative part of the dimension spectrum f() corresponding to the strongest singularities of the probability measure with 0. The right section of the spectrum, corresponding to the regular part of the measure, is not well defined for l0. These two effects are related to the fact that the generalized dimensions (q) exhibit (1) a first-order phase transition at q=1 and (2) a nonexistence of the thermodynamic limit l0, for q<0. We show that an appropriate description of the scaling can be obtained by considering the logarithm of the probability of picking a singularity on the fractal. The connections to fractal aggregates are briefly discussed. © 1994 The American Physical Society.