Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes

Abstract

Random coalescent processes and branching processes are two fundamental constructs in the field of stochastic processes, each with a rich history and a wide range of applications. Though developed within distinct contexts, in this note we present a novel connection between a multi-type (vector) multiplicative coalescent process and a multi-type branching process with Poisson offspring distributions. More specifically, we show that the equations that govern the phenomenon of gelation in the vector multiplicative coalescent process are equivalent to the set of equations that yield the extinction probabilities of the corresponding multi-type Poisson branching process. We then leverage this connection with two applications, one in each direction. The first is a new quick proof of gelation in the vector multiplicative coalescent process using a well known result of branching processes, and the second is a new series expression for the extinction probabilities of the multi-type Poisson branching process using results derived from the theory of vector multiplicative coalescence. While the correspondence is fairly straightforward, it illuminates a deep connection between these two paradigms which we hope will continue to reveal new insights and potential for cross-disciplinary research.