MULTIFRACTAL DIMENSIONS FOR BRANCHED GROWTH

A recently proposed theory for diffusion-limited aggregation (DLA), wh ich models this system as a random branched growth process, is reviewe d. Like DLA, this process is stochastic, and ensemble averaging is nee ded in order to define multifractal dimensions. In an earlier work by Halsey and Leibig, annealed average dimensions were computed for this model. In this paper, we compute the quenched average dimensions, whic h are expected to apply to typical members of the ensemble. We develop a perturbative expansion for the average of the logarithm of the mult ifractal partition function; the leading and subleading divergent term s in this expansion are then resummed to all orders. The result is tha t in the limit where the number of particles n --> infinity, the quenc hed and annealed dimensions are identical; however, the attainment of this limit requires enormous values of n. At smaller, more realistic v alues of n, the apparent quenched dimensions differ from the annealed dimensions. We interpret these results to mean that while multifractal ity as an ensemble property of random branched growth (and hence of DL A) is quite robust, it subtly fails for typical members of the ensembl e.