In the early eighties, physicists Belavin, Polyakov and Zamolodchikov postulated conformal invariance of critical planar statistical models. This prediction enabled physicists to harness 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 proved 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 (joint work with Smirnov) of a conjecture made by Nienhuis regarding the number of self-avoiding walks of length n on the hexagonal lattice starting at the origin.
org: GIULIANI Alessandro