Biased growth processes and the "rich-get-richer" principle

We study a simple stochastic system with a "rich-get-richer" behavior, in which there are 2 states, and N particles that are successively assigned to one of the states, with a probability p(i) that depends on the states' occupation n(i) as p(i) = n(gamma)(i)/ n(gamma)(1) + n(gamma)(2)). We show that there is a phase transition as gamma crosses the critical value gamma(c) =1. For gamma<1, in the thermodynamic limit the occupations are approximately the same, n(1) approximately n(2). For gamma>1, however, a spontaneous symmetry breaking occurs, and the system goes to a highly clustered configuration, in which one of the states has almost all the particles. These results also hold for any finite number of states (not only two). We show that this "rich-get-richer" principle governs the growth dynamics in a simple model of gravitational aggregation, and we argue that the same is true in all growth processes mediated by long-range forces like gravity.