Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator

Understand

Recently, two of us argued that the probability that an FK cluster in the Q-state Potts model connects three given points is related to the time-like Liouville three-point correlation function.

• Moreover, they predicted that the FK three-point connectivity has a prefactor which unveils the effects of a discrete symmetry, reminiscent of the S_Q permutation symmetry of the Q=2,3,4 Potts model.
• Their theoretical prediction has been checked for the case of percolation, corresponding to Q=1.
• We revisit the derivation of the time-like Liouville correlator given by Al.