The issues addressed in this project are related to the study of the asymptotic behaviour of certain stochastic processes. More precisely, the project analyzed the scaling limits of interacting particle systems. Mathematically speaking, these systems belong to the class of continuous-time Markov processes, whose dynamics conserve one or more quantities of interest (the thermodynamical quantities) and under a space-time scaling, the time evolution of these quantities can be described by distinct macroscopic laws. These laws can be partial differential equations or stochastic partial differential equations depending on whether one is looking at the convergence to the mean or at the fluctuations around that mean. Our aim was to give a step towards on the universality of these laws from microscopic stochastic systems.
In real-life situations, we come across many different episodes where we feel that it has similarities with something we have lived before in the past. Whether this is related to emotions or physical reactions, the similarities can be present. In nature, similarities between very different organisms also occur. As an example of this puzzling reality which somehow connects all of us, as an example we can recall a rigorous winter day in which ice particles fall from the sky. When seated in a car, we see a growing pattern of ice particles which is formed on the windscreen.
The ice particles fall, randomly, from the sky and when they hit the windscreen they form a growing pattern which can be seen in other, in principle, uncorrelated, situations such as coffee ring effects, bacterial growth like E-coli, the wake of flame, tumour growth…There are various physical systems, that when they are mathematically modelled they show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality. There are different shapes for these patterns and their study was the core of this project and it is related to a very active area of research in both mathematics and physics known as universality. This mysterious relationship between very distinct physical systems is encoded in some universal laws that one has to figure out how they can be characterized and how does one change from one universal law to another. They are linked by some parameter which somehow connects very different systems. This project analyzed this issue which is important for the understanding of the surrounding and mysterious world that we live in.
