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

TL;DR

This paper establishes a novel connection between vector multiplicative coalescent processes and multi-type Poisson branching processes, providing new proofs and expressions for gelation and extinction phenomena.

Contribution

It introduces a new equivalence between gelation equations and extinction probabilities, enabling simplified proofs and new series expressions for these processes.

Findings

New proof of gelation in vector multiplicative coalescent

Series expression for extinction probabilities of multi-type Poisson branching

Derivation of modified Smoluchowski equations using random graphs

Abstract

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 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, and the second is a new series expression for the extinction probabilities of the multi-type Poisson branching process. We also use random graphs to give a new derivation of the solution to the modified Smoluchowski coagulation equations, which describe the vector multiplicative coalescent process.