A Scaling Result For Explosive Processes

We consider the asymptotic behavior of the following model: balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n(p) for p > 1. In this case, one of the two bins eventually obtains a monopoly, in the sense that it obtains all balls thrown past some point. This model is motivated by the phenomenon of positive feedback, where the "rich get richer." We derive a simple asymptotic expression for the probability that bin 1 obtains a monopoly when bin 1 starts with x balls and bin 2 starts with y balls for the case f(n) = n(p). We then demonstrate the effectiveness of this approximation with some examples and demonstrate how it generalizes to a wide class of functions f.