**Lecture 1**

**Date and time: 9 January 2023, 15:30 - 16:30**

**Title**: Critical Phenomena Through the Lens of the Ising Model

**Abstract**: The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.

**Lectures 2-5**

**Dates and time: 10 & 11 January 2023, 11:15 - 12:05**

**Dates and time: 12 & 13 January 2023, 10:00 - 10:50**

**Title**: Marginal triviality of the scaling limits of critical 4D Ising

**Abstract**: We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

**About the speaker:**

Hugo Duminil-Copin received his PhD from the university of Geneva in 2012. He is permanent professor at the Institut des Hautes Etudes Scientifiques in France and full professor at the university of Geneva.

Hugo Duminil-Copin is a probabilist and a mathematical physicist. He works on statistical mechanics models such as percolation, the Ising model, the Potts model, random walks in random environments, random height functions. Hugo Duminil-Copin’s research has made contributions to percolation theory, a branch of probability theory that is concerned with the behavior of connected clusters in random graphs.His research also has an impact on mathematical physics, complex analysis, and combinatorics. In addition, he makes significant contributions to statistical physics.

Hugo Duminil-Copin received a number of awards, including the Fields medal in 2022.

This lecture series is part of the program "Topics in High Dimensional Probability".