Since Smoluchowski introduced his well-known coagulation equation in 1916, a substantial body of literature has developed to understand the properties of its solutions, as well as those of related stochastic models of coagulation. These models describe the evolution of a system of particles that merge pairwise over time, offering a microscopic, probabilistic perspective on the macroscopic dynamics captured by the Smoluchowski equation. In this talk, we will establish a connection between these particle systems and their deterministic limiting equation, thereby bridging the probabilistic and analytic points of view. Moreover, we will introduce the notion of a gelation-type phase transition and examine, as a representative example, the case of the multiplicative kernel. This kernel is notable for several reasons: it exhibits a gelation phase transition; it admits an explicit solution at the level of the Smoluchowski equation; and it has a one-to-one connection with the sparse Erdős–Rényi random graph.