Since Smoluchowski introduced his well-known coagulation equation in 1917, there has been an active line of research focused on understanding the properties of the solutions to this equation and related models for coagulation. In particular, in 2000, Norris introduced a generalised version of the model, which he named the cluster coagulation model. This model was intended to extend the framework established by Smoluchowski, allowing particles to have additional properties beyond their mass, such as shape or spatial location.

In this talk we focus on some recent progress in the study of the particle system that converges to such limiting (spatial) coagulation equation, often called the Marcus-Lushnikov process. In particular, we will present a recent sufficient criterion for the appearance of a giant particle (the gel in this framework) in the spatial setting. This improves existing criteria for gelation without the spatial interaction as well, proving in particular that homogeneous kernels with degree γ > 1 are indeed gelling (as long as they do not vanish on the diagonal). In addition, we present an approach based on Poisson Point Processes to study large deviations of the trajectory of such a Markov process in the large volume limit and explain how this also provides insight into gelation phenomena. This talk is based on a series of joint works with T. Iyer, W. König, H. Langhammer, E. Magnanini and R.I.A. Patterson.