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

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 (Delfino and Viti, 2011) [1]. Moreover, they predicted that the FK three-point connectivity has a prefactor which unveils the effects of a discrete symmetry, reminiscent of the SQ permutation symmetry of the Q=2,3,4 Potts model. We revisit the derivation of the time-like Liouville correlator (Zamolodchikov, 2005) [2] and show that this is 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.