Universality classes for strongly correlated models

• Applicant: Dr. Alexis Prévost
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552316285

The aim of this project is to rigorously study the phase transition of percolation models with strong correlations, and related universality classes. Percolation is a fundamental model of statistical physics which often plays a pivotal role to study the abrupt transition of various systems from a disordered phase to an ordered phase. Introducing strong correlations in these models entails studying physical objects with greater complexity than their independent counterparts, akin to analysing a random chain of mountains rather than independent peaks. Our goal is to rigorously quantify the speed of the phase transition and the geometry of the structures at the interface between the different phases using quantities called critical exponents. These exponents are notoriously challenging to compute rigorously, and even their very existence is often unclear. The proposed research programme investigates the values of these exponents for various percolation models, typically based on Gaussian fields or random walks. Our objective is to prove that these exponents depend only on the correlations of the model and the dimension of the underlying graph, indicating that they are part of the same universality class. Part of this programme is also dedicated to establishing links between these exponents and those of other well-known universality classes in statistical physics. Furthermore, the tools developed in this project will be used to investigate new properties of random walks on graphs.

• DFG Programme: Emmy Noether Independent Junior Research Groups