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Recently, two of us argued that the probability that an FK cluster in theQ-state Potts model connects three given points is related to the time-likeLiouville three-point correlation function. Moreover, they predicted that theFK three-point connectivity has a prefactor which unveils the effects of adiscrete symmetry, reminiscent of the S_Q permutation symmetry of the Q=2,3,4Potts model. Their theoretical prediction has been checked for the case ofpercolation, corresponding to Q=1. We revisit the derivation of the time-likeLiouville correlator given by Al. Zamolodchikov and show that this is the theonly consistent analytic continuation of the minimal model structure constants.We then present strong numerical tests of the relation between the time-likeLiouville correlator and percolative properties of the FK clusters for realvalues of Q.

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