First-passage times in a pure-birth coalescence model with size-dependent rates

We study a simple spatially homogeneous pure-birth growth model in which the size of a single “collector” cluster increases in unit steps with rate λn=knγ where n is the cluster size, and analyse the first-passage time (FPT) to reach a threshold size N. For arbitrary growth exponent γ>0 and initial size n0, we solve the discrete master equation and obtain an exact finite-N first passage time distribution (FPTD) in hypoexponential (phase-type) form via Laplace transforms. From this closed form, we derive explicit small-time and large-time asymptotics for fixed N, showing how the need to complete M=N−n0 sequential stages suppresses very early passages and how the slowest stage controls the far tail. We then analyse the large-N behaviour: for 0<γ≤1 the mean FPT diverges (as N1−γ or logN), while for γ>1 the FPT converges to a non-degenerate limit with finite Hurwitz-zeta moments and an infinite-product Laplace transform. In this strongly accelerating regime we obtain a small-time saddle-point asymptotic of essential-singularity type with an explicit power-law prefactor that makes the dependence on n0 and γ transparent. Together, these results clarify how N, γ, and n0 jointly shape the FPT distribution and provide a mathematically controlled benchmark linking exact stochastic growth to large-deviation onset criteria in applications such as warm-rain initiation and other aggregation-driven processes.