In this project we want to make serious innovative steps in the analysis of spatial random-particle systems undergoing coagulation and of the gelation phase transition, with the help of large deviation techniques and the analysis of the resulting Smoluchowski equation. In the first funding phase, we have been working in the hydrodynamic setting and laid foundations by deriving explicit formulas for the distribution of the particle configuration at a fixed time and large-deviation principles for the microscopic part of the configuration in the limit of a large system. In the second funding phase, we would like to extend these formulas to the macroscopic part and use them to gain a more detailed understanding of the role of space in the model. In particular, we will employ and extend a number of strategies for showing the occurrence of gelation for a wide class of spatial coagulation kernels. Additionally, we want to understand the phenomenon of mobility-induced gelation in a certain coagulation model with some random motion. Finally, we would like to make some non-trivial steps in the thermodynamic setting.

The desire to describe systems in which particles interact according to chemical reaction mechanisms has led to a range of challenging mathematical models. This project focuses on one of the most fundamental: particle growth due to coagulation. We want to make serious innovative steps in the analysis of spatial random-particle models for coagulation and the gelation phase transition, with the help of techniques from the mathematical theory of large-deviations.