FK-Percolation and the Six-Vertex Model: Critical Phenomena

This lecture discusses the transition from the six-vertex model to FK-percolation and back, focusing on critical phenomena in two-dimensional systems. It begins with an overview of monotonicity properties for FK-percolation, emphasizing the FKG inequality and the thermodynamic limit. The instructor presents duality and auto-duality concepts in two dimensions, highlighting their implications for phase transitions. The lecture further explores the dichotomy of critical FK-percolation, detailing conditions under which unique Gibbs measures exist and the behavior of infinite clusters. The continuity and discontinuity of phase transitions are examined, with references to recent research. The instructor introduces conjectures regarding the scaling limit of critical FK-percolation, discussing the fractal behavior of cluster boundaries and the significance of conformally invariant scaling limits. The lecture concludes with a discussion on loop representation and topology, emphasizing the distance between loops and the measures on families of loops, providing a comprehensive understanding of the interplay between these mathematical concepts.