For 8/3 < κ < 8, the conformal loop ensemble CLE κ is a canonical random ensemble of loops which is conformally invariant in law, and whose loops locally look like Schramm-Loewner evolution with parameter κ. It describes the scaling limits of the Ising model, percolation, and other models. When κ ≤ 4 the loops are simple curves. In this regime we compute the three-point function of CLE κ on the sphere, and show it agrees with the imaginary DOZZ formula of Zamolodchikov (2005). We also verify a conjecture of Kenyon and Wilson on the electrical thickness of CLE κ on the sphere. Our arguments depend on couplings of CLE with Liouville quantum gravity and the integrability of Liouville conformal field theory.
Based on joint work with Xin Sun, which builds on our recent work with Holden and Remy.