# Bilinear Coagulation Equations

## TL;DR

This paper studies bilinear coagulation equations, establishing a phase transition at gelation time, and connects the stochastic process to random graph models to analyze behavior before and after gelation.

## Contribution

It introduces a general class of coagulation kernels with bilinear form, characterizes the gelation transition via eigenvalue problems, and extends analysis using random graph couplings.

## Key findings

- Gelation occurs at a finite time characterized by an eigenvalue problem.
- Hydrodynamic limit established for stochastic coagulation process.
- Analysis extended beyond gelation time through random graph coupling.

## Abstract

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form $\pi(y)\cdot A\pi(x)$ for a vector of conserved quantities $\pi$, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time $t_\mathrm{g}\in (0,\infty)$, which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.