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Stochastic Coagulation and the Timescale for Runaway Growth

We study the stochastic coagulation equation using simplified models and efficient Monte Carlo simulations. It is known that (i) runaway growth occurs if the two-body coalescence kernel rises faster than linearly in the mass of the heavier particle; and (ii) for such kernels, runaway is instantaneous in the limit that the number of particles tends to infinity at fixed collision time per particle. Superlinear kernels arise in astrophysical systems where gravitational focusing is important, such as the coalescence of planetesimals to form planets or of stars to form supermassive black holes. We find that the time required for runaway decreases as a power of the logarithm of the the initial number of particles. Astrophysical implications are briefly discussed.