Preferential growth: Exact solution of the time-dependent distributions - art. no. 051112

We consider a preferential growth model where particles are added one by on e to the system consisting of clusters of particles. A new particle can eit her form a new cluster (with probability q) or join an already existing clu ster with a probability proportional to the size thereof. We calculate exac tly the probability P-i(k,t) that the size of the ith cluster at time t is k. We analyze the asymptotics, the scaling properties of the size distribut ion and of the mean size, as well as the relation of our system to recent n etwork models.