Stochastic multifractality and universal scaling distributions

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Abstract
A hierarchical random process can be characterized by a scaling probability distribution describing the stochastic multiplicative process occurring at each generation. Replica averages of the associated random partition functions yield multifractal spectra τ(q,n) dependent on the number of members n in the replica average chosen, with the exponents generated by quenched (n=0) and annealed (n=1) averaging being but two points in this continuous spectrum. As a consequence the α versus f(α) description has to be generalized for stochastic multifractals. Robust properties of these multifractal spectra, including the position and singularities of generalized dimensions at phase transitions, spectral inequalities and asymptotics, and the scaling behavior of minimal probabilities, are found to depend on universal properties of the scaling probability distribution, specifically on the form of singularities in the distribution, the strength of correlations between measure and length scale during fragmentation, and whether or not the process is conservative. We apply our approach to both the mass and growth scaling probability distributions for diffusion-limited aggregation and show that their construction is Markovian, but while the mass measure is log normal, the singularities in the growth measure do not obey the central limit theorem.

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