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This lecture explores the integrability of Conformal Loop Ensemble (CLE) and its connection to the imaginary DOZZ formula. Discover how the scaling limit of probability for points in the same cluster in 2D critical percolation relates to conformal field theory. Learn about Delfino and Viti's groundbreaking prediction on three-point probability expressed through the imaginary DOZZ formula, and how similar conjectures were developed for scaling limits of random cluster models and O(n) loop models. Understand how these mathematical structures combine integrable structures of discrete models with bootstrap hypothesis to represent three-point observables. The presentation demonstrates how these conjectures can be formulated as exact statements on CLE observables and explains the derivation of three-point functions through Liouville quantum gravity. The talk is based on joint work with Morris Ang (UC San Diego), Gefei Cai (BICMR), and Xin Sun (BICMR).
