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We revisit in this paper the problem of connectivity correlations in theFortuin-Kasteleyn cluster representation of the two-dimensional $Q$-state Pottsmodel conformal field theory. In a recent work [M. Picco, S. Ribault and R.Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], results for thefour-point functions were obtained, based on the bootstrap approach, combinedwith simple conjectures for the spectra in the different fusion channels. Inthis paper, we test these conjectures using lattice algebraic considerationscombined with extensive numerical studies of correlations on infinitecylinders. We find that the spectra in the scaling limit are much richer thanthose proposed in [arXiv:1607.07224]: they involve in particular fields withconformal weight $h_{r,s}$ where $r$ is dense on the real axis.

Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra