Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes
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.
Keywords
Cite
@article{arxiv.2409.06910,
title = {Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes},
author = {Heshan Aravinda and Yevgeniy Kovchegov and Peter T. Otto and Amites Sarkar},
journal= {arXiv preprint arXiv:2409.06910},
year = {2025}
}
Comments
13 pages, 2 figures, 1 table; The last section has been expanded to include additional details on the conceptual connection between coalescence, random graphs and branching processes