Description
This series of lectures introduces stochastic models for coagulation and their connections with kinetic equations, random graphs, and phase transitions. We begin with the classical Marcus–Lushnikov process, a finite-particle Markov model in which clusters merge at rates prescribed by a coagulation kernel, and show how the Smoluchowski coagulation equation emerges as its law-of-large-numbers limit. Special attention will be given to some kernels which are exactly solvable, and will serve as guiding examples throughout the course. In particular, the multiplicative kernel which provides a fundamental bridge between coagulation theory and random graph processes.
A central theme will be the phenomenon of gelation, where mass is lost from the limiting kinetic equation and, at the particle level, macroscopic clusters appear in finite time. We discuss how this phase transition can be understood both analytically, through moment estimates and mass conservation, and probabilistically.
The lectures will also explore extensions beyond the classical space homogeneous setting, including cluster-valued, inhomogeneous, and spatial coagulation models. Along the way, we emphasize the probabilistic tools used to study these systems: martingale formulations, hydrodynamic limits, random graphs, with a special focus on large-deviation estimates. The goal is to provide a unified introduction to stochastic coagulation models, from foundational mean-field limits to current research questions on gelation, criticality, and spatial inhomogeneity.