Expected Coalescence Time for a Nonuniform Allocation Process

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 p 2 j , the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c, under two constraints on p that exclude a thin set of distributions p. One of the constraints is c ≪ ln n. This ln n is shown to be a threshold value: for c ≫ ln n, there exists p with c(p) = c such that the expected coalescence time far exceeds c. Connections to coalescent processes in population biology and theoretical computer science are discussed. AMS 2000 Subject classifications. Primary 60C05; secondary 05A15, 60F10, 60J05, 65K10, 68W40, 92D25.