Abstract: In the context of random cluster models, the connectivity functions denoted as $P_n(x_1, x_2, ..., x_n)$ signify the probabilities associated with n points belonging to the same finite cluster. The initial conjecture by Delfino and Viti proposed that, at the critical point in the continuum limit, the ratio $R = P_3(x_1, x_2, x_3) / \sqrt{P_2(x_1, x_2) P_2(x_2, x_3) P_3(x_1, x_3)}$ converges to a universal constant solely dependent on $\kappa$. This dependence can be expressed through the imaginary DOZZ formula. For percolation, this constant approximates to 1.022. In this presentation, we elucidate the proof specifically for the percolation scenario. Additionally, we introduce analogous quantities within the conformal loop ensembles carpet/gasket measure, demonstrating their precise alignment with the imaginary DOZZ formula. The discussion will also delve into the statistical physics origin and its connections to conformal field theory.
This is based on the joint work with Morris Ang (Columbia), Gefei Cai (BICMR), and Xin Sun (BICMR).