Exact Calculation of the Spatio-temporal Correlations in the Takayasu model and in the q-model of Force Fluctuations in Bead Packs

We calculate exactly the two point mass-mass correlations in arbitrary spatial dimensions in the aggregation model of Takayasu. In this model, masses diffuse on a lattice, coalesce upon contact and adsorb unit mass from outside at a constant rate. Our exact calculation of the variance of mass at a given site proves explicitly, without making any assumption of scaling, that the upper critical dimension of the model is 2. We also extend our method to calculate the spatio-temporal correlations in a generalized class of models with aggregation, fragmentation and injection which include, in particular, the $q$-model of force fluctuations in bead packs. We present explicit expressions for the spatio-temporal force-force correlation function in the $q$-model. These can be used to test the applicability of the $q$-model in experiments.