Abstract:
Coagulation models were first formulated in the early twentieth century in colloidal chemistry. They entered modern discrete probability through the work of Aldous, who revealed a striking connection between the multiplicative coalescent and the Erdős–Rényi random graph in the critical regime. This connection gives a precise mathematical description of how macroscopic components, or “gels,” emerge from the coagulation of microscopic dust.
The aim of this talk is to illustrate the two-way flow of ideas between coagulation models and random graph theory. On one hand, random graphs provide canonical examples of coagulation dynamics in which clusters merge through random connections. On the other hand, the coagulation perspective has become a powerful framework for proving results in modern probabilistic combinatorics. I will describe applications including:
- universality of the critical regime for broad classes of random graph models;
- scaling limits for random edge-weighted minimal spanning trees and related Erdős–Rényi graph processes;
- functional central limit theorems for both microscopic and macroscopic observables.
If time permits, I will also discuss recent connections between multiplicative coalescents with immigration and network archaeology for growing networks.