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Expected Coalescence Time for a Nonuniform Allocation Process

By John K. Mcsweeney and Boris G. Pittel


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: = P j p2j, 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 ≪ ln −2 n. This ln −2 n is shown to be a threshold value: for c ≫ 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

Topics: Fisher model, random functions, Markov chain, asymptotic behavior
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