Dmitry Krachun (Princeton): Towards conformal invariance of critical planar lattice models

Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the proof of rotational invariance of the FK-percolation model and more recently the computation of critical exponents of planar FK model with q=4. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz. We will then explore the connection to the six-vertex model further to derive critical exponents of the FK model with q=4.

Based on joint works with Hong-Bin Chen, Hugo Duminil-Copin, Tiancheng He, François Jacopin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, Tatiana Tikhonovskaia, and Jiaming Xia.