Le 8ème Séminaire Itzykson, portant sur le thème "Parafermionic observables and 2D statistical physics" s'est tenu à l'IHÉS (plan d'accès) le Lundi 22 janvier 2018.
10h-12h Hugo Duminil-Copin (IHES)
Introduction to parafermionic observables
In the early eighties, the physicists Belavin, Polyakov and Zamolodchikov postulated the conformal invariance of critical planar statistical models. This prediction enabled physicists to use Conformal Field Theory in order to formulate many conjectures on these models. From a mathematical perspective, proving rigorously the conformal invariance of a model (and properties following from it) constitutes a formidable challenge. In recent years, the connection between discrete holomorphicity and planar statistical physics led to spectacular progress in this direction. Kenyon, Chelkak and Smirnov exhibited discrete holomorphic observables in the dimer and Ising models and roved their convergence to conformal maps in the scaling limit. These results paved the way to the rigorous proof of conformal invariance for these two models.
Other discrete observables have been proposed for a number of critical models, including self-avoiding walks and Potts models. While these observables are not exactly discrete holomorphic, their discrete contour integrals vanish, a property shared by discrete holomorphic functions. This property sheds a new light on the critical models, and we propose to discuss some of its applications. In particular, we will sketch the proof of a conjecture made by Nienhuis regarding the number of self-avoiding walks of length n on the hexagonal lattice.
14h-15h: Denis Bernard (ENS Paris)
A (biased) semi-historical introduction to parafermionic fields in statistical physics
Starting from Kadanoff's and Ceva's construction, I will present an elementary introduction to different reincarnations of parafermionic observables and fields in statistical models and their relationships with braid group representations, quantum group symmetries, integrability and holomorphicity.
15h15-16h15: Dmitry Chelkak (ENS Paris & PDMI, St.Petersburg)
2D Ising fermions: from combinatorics to conformal invariance and s-embeddings
During the last decade, a number of results on conformal invariance of the critical 2D nearest-neighbor Ising model appeared, both for the small mesh size limits of correlation functions (fermions, energy densities, spins, disorders etc) and interfaces (aka domain walls) or loop ensembles arising in the model. We survey these results, which are based on discrete analysis techniques, notably on convergence theorems for solutions to some special boundary value problems for discrete holomorphic functions on rhombic lattices, and discuss the universality of the model with respect to an underlying planar graph.