Public presentation of Prof. Ioan Manolescu of the Department of Mathematics. The conference is given in the context of the promotion commission

Towards identifying the scaling limit of critical FK-percolation -

Abstract:
The related notions of phase transition and universality are central to statistical mechanics, and may be illustrated through the probabilistic study of lattice models. Among them is the random-cluster model (or Fortuin-Kasteleyn percolation), of which percolation is a particular instance. It is a model of random connections in a graph that exhibits a phase transition expressed in terms of the existence of infinite connected components. In this talk I aim to present two major results about the phase transition of the planar model obtained in the last year.
When considered at its critical point, the random-cluster model is expected to exhibit a scaling limit which is universal and invariant under a large class of transformations of the plane, known as conformal maps. In particular, its large scale behaviour is expected to be invariant under rotations. We prove this, along with a universality result, using the star-triangle transformation.
Critical exponents govern the decrease of certain critical quantities, and thus characterise the phase transition. A large family of critical exponents may be associated with any statistical mechanics model, but they are expected to be related to each other via certain scaling relations. We prove these scaling relations for the random-cluster model, thus generalising the celebrated result of Kesten for percolation (1987) and illustrating their universality.


Depending on the COVID evolution, the conference might be entirely online or in hybrid format.