Abstract
The spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated. The model is characterized byrate constants K ij =i+j for the clustering of ani- and aj-mer, anddiffusion constants D j =D. It is assumed thatD≫1 (reaction-limited aggregation). Explicit expressions for the correlation functions at equal and at different times are calculated. The equal-time correlation functions show scaling behavior in the scaling limit. The correlation length remains finite ast→∞, and the fluctuations becomelarge at large times (t⩾t D ) inany dimension. The crossover timet D , at which the mean field theory (Smoluchowski's equation) breaks down, is given byt D ≃InD. These exact results imply that the upper critical dimension of this model isd c=∞ and, hence, that there isno unique upper critical dimension in models for the irreversible aggregation of clusters.
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