Multifractal Dimensions for Branched Growth

TL;DR

This paper compares quenched and annealed multifractal dimensions in a stochastic model of diffusion-limited aggregation, revealing that differences diminish at large particle numbers but are significant at realistic sizes.

Contribution

It introduces a perturbative expansion to compute quenched multifractal dimensions and shows their convergence to annealed dimensions at large ensemble sizes.

Findings

Quenched and annealed dimensions are identical as particle number n approaches infinity.

At realistic particle numbers, quenched and annealed dimensions differ.

Multifractality is robust as an ensemble property but can subtly fail for typical members.

Abstract

A recently proposed theory for diffusion-limited aggregation (DLA), which models this system as a random branched growth process, is reviewed. Like DLA, this process is stochastic, and ensemble averaging is needed in order to define multifractal dimensions. In an earlier work [T. C. Halsey and M. Leibig, Phys. Rev. A46, 7793 (1992)], annealed average dimensions were computed for this model. In this paper, we compute the quenched average dimensions, which are expected to apply to typical members of the ensemble. We develop a perturbative expansion for the average of the logarithm of the multifractal partition function; the leading and sub-leading divergent terms in this expansion are then resummed to all orders. The result is that in the limit where the number of particles n -> \infty, the quenched and annealed dimensions are {\it identical}; however, the attainment of this limit requires enormous values of n. At smaller, more realistic values of n, the apparent quenched dimensions differ from the annealed dimensions. We interpret these results to mean that while multifractality as an ensemble property of random branched growth (and hence of DLA) is quite robust, it subtly fails for typical members of the ensemble.