Patrícia Gonçalves received her PhD in 2007 from IMPA, Brazil. She is currently a full Professor at Instituto Superior Técnico, University of Lisbon. She works in the field of interacting particle systems and her work has particularly focused on their scaling limits, namely on hydrodynamic limits, fluctuations and large deviations. Patrícia is the editor-in-chief of the Electronic Journal of Probability and associate editor of Annals of Probability. She has also been on the editorial board of the Annals of Applied Probability and Electronic Journal of Probability. In 2016, she was the recipient of an ERC Starting Grant in Mathematics and in 2022 was an invited speaker at the sectional session in Probability at the International Congress of Mathematicians. This Schramm lecture will be delivered at the 11th World Congress in Probability and Statistics in Bochum, Germany, August 12–16, 2024: https://www.bernoulli-ims-worldcongress2024.org

Scaling limits of general exclusion processes

The rigorous mathematical derivation of the macroscopic evolution equations of classical fluid mechanics from the large-scale description of the conserved quantities in Newtonian particle systems is a long-standing problem in mathematical physics. Instead, if the deterministic dynamics are replaced by stochastic dynamics, then the mathematical techniques that have been developed in the last decades can be efficient. Over the last 30 years, there has been remarkable progress in deriving the well-known hydrodynamic limit, from stochastic interacting particle systems, as well as, characterizing the fluctuations of locally conserved quantities around that limit.

The goal of my talk is to review old and recent results about these scaling limits with a special focus on exclusion processes. In this process, each site can be occupied at most by one particle and after an exponential clock of rate 1, it jumps to another position of a lattice according to a certain transition probability. Depending on the transition probability different limit behaviours can be obtained and the system can cross different universality classes. These crossovers will be explained both at the level of hydrodynamic limits as well as fluctuations.