A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millennium, the mathematical understanding of this fact progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models.
In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. Percolation models are models of random subgraphs of a given lattice. They have a rich history and lie at the crossroad of several families of lattice models, in particular in two dimensions. In recent years, the understanding of their critical behaviour progressed greatly in the planar case.
We will explain some recent proof of rotational invariance for a large class of such percolation models, called the random-cluster models or Fortuin-Kasteleyn percolation. This represents an important progress in the direction of proving full conformal invariance. We will also explain what are the missing ingredients to prove the full conformal invariance.
This is based on joint work with Karol Kozlowski, Dmitry Krachun, Ioan Manolescu, and Mendes Oulamara.
Hugo Duminil-Copin studied at the Lycée Louis-le-Grand in Paris, then the University Paris-Sud and the École normale supérieure (Paris). In 2008, he moved to the University of Geneva to write a PhD thesis under Stanislav Smirnov. In 2013, after his postdoctorate, he was appointed assistant professor, then professor, at the University of Geneva. In 2016, he became permanent professor at the Institut des Hautes Études Scientifiques.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai