**Time:** 14:00-15:00, Monday, June 24 2024

**Venue:** E4-233, Yungu Campus & ZOOM

**ZOOM ID: **984 0500 2228

**PASSCODE:** 268867

**Host: **Weronika Czerniawska, ITS

**Speaker:** Dmitry Krachun, Princeton University

**Biography: **Dmitry Krachun completed his PhD under the supervision of Hugo Dumini-Copin (Fields Medal 2010) at Geneva University in 2023. He has been conducting research in statistical mechanics. In particular, using exact integrability techniques to shed light on the fine properties of the critical random cluster model in two dimensions and proving its rotational invariance at large scales. This was a joint project with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, and Mendes Oulamara. In September 2023 he joined the mathematics department of Princeton University.

**Title: **A glimpse of universality in critical planar models

**Abstract: **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 recent result which proves rotational invariance of the FK-percolation model. 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. Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.