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

Heshan Aravinda, Yevgeniy Kovchegov, Peter T. Otto, Amites Sarkar

TL;DR

The paper uncovers a fundamental link between the vector-multiplicative coalescent and multi-type Poisson branching processes, showing that gelation dynamics correspond to extinction probabilities under a shared Lambert-Euler framework. It derives a quick gelation proof from this correspondence and provides a novel series representation for extinction probabilities, plus a random-graph derivation of the pre-gelation Smoluchowski solution. The results unify coalescence, branching, and random-graph formalisms, offering cross-disciplinary insights and computable formulas via weighted spanning-tree enumerators and Lambert-W-type functions. The work also clarifies the role of the interaction matrix $V$ and initial masses $\boldsymbol{\alpha}$ in governing gelation time and extinction behavior.

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.

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

TL;DR

The paper uncovers a fundamental link between the vector-multiplicative coalescent and multi-type Poisson branching processes, showing that gelation dynamics correspond to extinction probabilities under a shared Lambert-Euler framework. It derives a quick gelation proof from this correspondence and provides a novel series representation for extinction probabilities, plus a random-graph derivation of the pre-gelation Smoluchowski solution. The results unify coalescence, branching, and random-graph formalisms, offering cross-disciplinary insights and computable formulas via weighted spanning-tree enumerators and Lambert-W-type functions. The work also clarifies the role of the interaction matrix and initial masses in governing gelation time and extinction behavior.

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.
Paper Structure (8 sections, 5 theorems, 46 equations, 2 figures, 1 table)

This paper contains 8 sections, 5 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vector multiplicative coalescent processes
  3. The solution to the Smoluchowski equations prior to gelation
  4. Random graphs and an alternative proof of equation \ref{['eqn:MSEsol']}
  5. Gelation and Lamber-Euler Inversion
  6. Multi-Type Poisson Branching Processes
  7. Extinction Probability
  8. Correspondence between Gelation and Extinction Probability with Applications

Key Result

Theorem 2.1

KO For given ${\boldsymbol\alpha},V$ and for any ${\bf x}$ and $t<t'=T_{\rm gel}$, we have where the functions $\zeta_{\bf x}(t)$ are the solutions of the modified Smoluchowski equations eqn:VMSmolEqn above.

Figures (2)

  • Figure 1: Post-gelation behavior of Lambert-Euler inversion
  • Figure 2: Three coupled models: $C$, $G$ and $B$

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 3.1: Multidimensional Lambert-Euler inversion
  • Theorem 4.1
  • Theorem 5.1
  • proof
  • Remark 5.2
  • Theorem 5.3