This semester the Amsterdam Math-Physics Colloquium will be on May 22 Tuesday, and the speakers will be Prof. Hugo Duminil (IHÉS) and Prof. Jesper Jacobsen (ENS Paris). The schedule, titles and abstracts can be found below.
13.30-14.00 Coffee and Tea
14.00-15.00 talk by Prof. H. Duminil (IHÉS)
15.00-15.30 Coffee and Tea
15.30-16.30 talk by Prof. J. Jacobsen (ENS Paris)
16.30 - Drinks
Title: Counting self-avoiding paths using discrete holomorphic functions
Abstract: 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 for 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.
Title: “Four-point functions in the Fortuin-Kasteleyn cluster model”.
Abstract: The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model.