Phase Transitions and Scaling Limits

Statistical physics provides a rich mathematical framework for understanding large systems of interacting particles, and in particular the universal phenomena of phase transitions. These transitions abrupt changes in the macroscopic behavior of a system play a central role not only in physics, but also in areas such as biology, computer science, and social dynamics. The project aims to reveal a unifying probabilistic structure of these transitions, including symmetries emerging at the scaling limit. In 2000, Schramm revolutionized the field by formulating a conjecture for the scaling limit of interfaces in two dimensions. Since then, this has become a major research topic in mathematics, with Fields Medals awarded to Werner (2006), Smirnov (2010), and Duminil- Copin (2022). The Nobel Prize in Physics in 2016 was awarded to Kosterlitz and Thouless for the discovery of topological infinite-order phase transitions, also studied in this project. Building on these breakthroughs, the project seeks to deepen our understanding of two- dimensional critical systems, focusing on the geometry of interfaces, emergent symmetries, and the universal structures that govern them. A guiding idea is the existence of a universal monotonic structure, expressed through graphical representations of interactions, that underlies a broad class of models. Two central models are the Ising model and percolation. The Ising model of ferromagnetism, introduced over 100 years ago, is the most studied in statistical physics. Percolation theory describes the random formation of clusters in a network and serves as a paradigm of geometric phase transitions and a testing ground for universality and scaling. Their interplay with more general spin and loop systems is at the heart of the project. The analysis relies on probabilistic tools such as positive correlation (FKG) inequalities, which imply monotonicity and allow one to establish sharp phase transitions and decouple complex dependent systems. These methods are combined with tools of exact integrability (Yang-Baxter or star-triangle transformations) and discrete complex analysis (Smirnovs holomorphic observables) to reveal emergent symmetries in the scaling limit. These symmetries are absent at a discrete microscopic level but govern the macroscopic behavior. Beyond classical lattice models, the project also investigates systems on general planar graphs, with connections to quantum spin chains. By studying a wide range of models, the project aims to develop a robust and unified approach to phase transitions and scaling limits, uncovering universal mechanisms that shape critical phenomena across mathematics, physics, and beyond.