# Probability and Mathematical Physics Seminar

#### A glimpse of universality in critical planar lattice models

**Speaker:**
Dmitrii Krachun, Princeton University

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, March 1, 2024, 11:10 a.m.

**Synopsis:**

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.