Statistical physics is a theory allowing the derivation of the statistical behaviour of macroscopic systems from the description of the interactions of their microscopic constituents. For
more than a century, lattice models (i.e. random systems defined on lattices) have been introduced as discrete models describing the phase transition for a large variety of phenomena, ranging from ferroelectrics to lattice gas. In the last decades, our understanding of percolation and the Ising model, two classical examples of lattice models, progressed greatly. Nonetheless, major questions remain open on these two models. The goal of this project is to break new grounds in the understanding of phase transition in statistical physics by using and aggregating in a pioneering way multiple techniques from probability, combinatorics, analysis and integrable systems.
In this project, we focus on three main goals:
- provide a solid mathematical framework for the study of universality for Bernoulli percolation and the Ising model in two dimensions:
- Advance in the understanding of the critical behaviour of Bernoulli percolation and the Ising model in dimensions larger or equal to 3.
- Greatly improve the understanding of planar lattice models obtained by generalizations of percolation and the Ising model, through the design of an innovative mathematical theory of phase transition dedicated to graphical representations of classical lattice models, such as Fortuin-Kasteleyn percolation, Ashkin-Teller models and Loop models.
Most of the questions that we propose to tackle are notoriously difficult open problems whose solution will change our understanding of phase transitions.
Since phase transitions in mathematics are related to the phase transition of physical systems in our everyday life (paramagnetic/ferromagnetic phase transition, interfaces in erosion models, forest fires, etc), any progress in their understanding will shed a new light on the phase transition of these physical systems and will potentially have applications beyond mathematics and mathematical physics.
