Probability and Mathematical Physics Seminar

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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.