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Expected coalescence time for a nonuniform allocation process

  • John K. McSweeney (a1) and Boris G. Pittel (a1)

Abstract

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p . We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑ j p j 2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c −1, under two constraints on p that exclude a thin set of distributions p . One of the constraints is c = o(ln−2 n). This ln−2 n is shown to be a threshold value: for c = ω(ln−2 n), there exists p with c( p ) = c such that the expected coalescence time far exceeds c −1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

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Copyright

Corresponding author

Postal address: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA.
∗∗ Email address: mcsweeney@math.ohio-state.edu
∗∗∗ Email address: bgp@math.ohio-state.edu

References

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Keywords

MSC classification

Expected coalescence time for a nonuniform allocation process

  • John K. McSweeney (a1) and Boris G. Pittel (a1)

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