Speaker
Description
The scaling limit of the probability that n points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory ($CFT$). Although this is not fully understood, Delfino and Viti made a remarkable prediction on the exact value of a properly normalized three-point probability from the exact $S$-matrix. It is expressed in terms of the imaginary $DOZZ$ formula. Later, similar conjectures were made for scaling limits of random cluster models and $O(n)$ loop models, combining both integrable structure of discrete model as well as the bootstrap hypothesis, representing certain three-point observables in terms of the imaginary $DOZZ$ formula and its variants. Since the scaling limits of these models can be described by the conformal loop ensemble ($CLE$), such conjectures can be formulated as exact statements on $CLE$ observables. This talk explains the derivation of the above three-point functions via Liouville quantum gravity.
This is based on the joint work with Morris Ang (UC San Diego), Gefei Cai (BICMR), and Xin Sun (BICMR).