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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 $f$.