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

Abstract: 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. Zamolodchikov and show that this is the the only consistent analytic continuation of the minimal model structure constants. We then present strong numerical tests of the relation between the time-like Liouville correlator and percolative properties of the FK clusters for real values of Q.