Random fractals, phase transitions, and negative dimension spectra

Abstract

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(\ensuremath{\alpha}) corresponding to the strongest singularities of the probability measure with \ensuremath{\alpha}\ensuremath{\approxeq}0. The right section of the spectrum, corresponding to the regular part of the measure, is not well defined for l\ensuremath{\rightarrow}0. These two effects are related to the fact that the generalized dimensions \ensuremath{\tau}(q) exhibit (1) a first-order phase transition at q=1 and (2) a nonexistence of the thermodynamic limit l\ensuremath{\rightarrow}0, for q0. We show that an appropriate description of the scaling can be obtained by considering the logarithm of the probability of picking a singularity \ensuremath{\alpha} on the fractal. The connections to fractal aggregates are briefly discussed.