Pauli Lectures 2025

With an example borrowed from the research of Professor Hugo Duminil-Copin, we will illustrate how counting can shed light on the behavior of complex physical systems, while simultaneously revealing the need to sometimes go beyond what numbers tell us in order to unveil all the mysteries of the world around us.

Phase transitions mark sudden, dramatic changes in the behavior of complex systems. To explore the mathematics behind these phenomena, we will focus on two of the most fundamental models in statistical physics: the Ising model and percolation. Originally introduced to capture ferromagnetism and the flow through porous media, these models have since evolved into a rich mathematical framework and a versatile tool for understanding abrupt shifts across diverse settings. In this talk, I will survey key advances in their study and highlight the broader insights they offer into the nature of phase transitions in statistical physics.

In this talk, we will explore the rich interplay between two-dimensional critical percolation models and the six-vertex model, a classical integrable random height model. By leveraging the remarkable symmetries and emergent structures that arise in the large-scale behavior of these systems, we will discuss how the so-called Coulomb Gas Formalism may be placed on rigorous mathematical foundations in this context. This perspective opens new pathways toward a deeper mathematical understanding of the phase transition of these models. The presentation is intended to be accessible to a broad mathematical audience.