A scaling result for explosive processes
Summary: We consider the asymptotic behavior of the following model: balls are sequentially throw 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 \(p>14\). 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 \(f\).
scientific article; zbMATH DE number 2107473