A recently proposed theory for diffusion-limited aggregation (DLA), whichmodels this system as a random branched growth process, is reviewed. Like DLA,this process is stochastic, and ensemble averaging is needed in order to definemultifractal dimensions. In an earlier work [T. C. Halsey and M. Leibig, Phys.Rev. A46, 7793 (1992)], annealed average dimensions were computed for thismodel. In this paper, we compute the quenched average dimensions, which areexpected to apply to typical members of the ensemble. We develop a perturbativeexpansion for the average of the logarithm of the multifractal partitionfunction; the leading and sub-leading divergent terms in this expansion arethen resummed to all orders. The result is that in the limit where the numberof particles n -> \infty, the quenched and annealed dimensions are {\itidentical}; however, the attainment of this limit requires enormous values ofn. At smaller, more realistic values of n, the apparent quenched dimensionsdiffer from the annealed dimensions. We interpret these results to mean thatwhile multifractality as an ensemble property of random branched growth (andhence of DLA) is quite robust, it subtly fails for typical members of theensemble.
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机译：审查了最近提出的扩散受限聚集（DLA）理论，该理论将该系统建模为随机分支生长过程。像DLA一样，此过程是随机的，需要集成平均以定义多重分形维数。在更早的工作中[T. C. Halsey和M. Leibig，物理版。 A46，7793（1992）]，计算了该模型的退火平均尺寸。在本文中，我们计算了淬火平均尺寸，该尺寸预计将应用于典型的整体成员。我们为多重分形分配函数的对数的平均值开发了一个摄动展开式；然后将此扩展中的领导和次领导分歧条款恢复为所有订单。结果是，在n-> \ infty的粒子数的极限中，淬火和退火的尺寸为{\ itidentical}；但是，要达到此极限，需要巨大的n值。在较小，更实际的n值下，表观淬火尺寸与退火尺寸不同。我们将这些结果解释为意味着，尽管多重分形作为随机分支增长的集合性质（DLA的存在）非常稳健，但对于该集合的典型成员却微不足道。
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