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We study transitions in log-correlated Random Energy Models (logREMs) thatare related to the violation of a Seiberg bound in Liouville field theory(LFT): the binding transition and the termination point transition (a.k.a.pre-freezing). By means of LFT-logREM mapping, replica symmetry breaking andtraveling-wave equation techniques, we unify both transitions in atwo-parameter diagram, which describes the free energy large deviations oflogREMs with a deterministic background log potential, or equivalently, thejoint moments of the free energy and Gibbs measure in logREMs withoutbackground potential. Under the LFT-logREM mapping, the transitions correspondto the competition of discrete and continuous terms in a four-point correlationfunction. Our results provide a statistical interpretation of a peculiarnon-locality of the operator product expansion in LFT. The results arere-derived by a traveling-wave equation calculation, which shows that thefeatures of LFT responsible for the transitions are reproduced in a simplemodel of diffusion with absorption. We examine also the problem by a replicasymmetry breaking analysis. It complements the previous methods and reveals arich large deviation structure of the free energy of logREMs with adeterministic background log potential. Many results are verified in theintegrable circular logREM, by a replica-Coulomb gas integral approach. Therelated problem of common length (overlap) distribution is also considered. Weprovide a traveling-wave equation derivation of the LFT predictions announcedin a precedent work.

Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models