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

Xiangyu CaoPierre Le Doussal 1 Alberto Rosso 2 Raoul Santachiara 2

Phys.Rev.E, 2018, 97 (4), pp.042111. 〈10.1103/PhysRevE.97.042111〉

We study transitions in log-correlated random energy models (logREMs) that are 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 and traveling-wave equation techniques, we unify both transitions in a two-parameter diagram, which describes the free-energy large deviations of logREMs with a deterministic background log potential, or equivalently, the joint moments of the free energy and Gibbs measure in logREMs without background potential. Under the LFT-logREM mapping, the transitions correspond to the competition of discrete and continuous terms in a four-point correlation function. Our results provide a statistical interpretation of a peculiar nonlocality of the operator product expansion in LFT. The results are rederived by a traveling-wave equation calculation, which shows that the features of LFT responsible for the transitions are reproduced in a simple model of diffusion with absorption. We examine also the problem by a replica symmetry breaking analysis. It complements the previous methods and reveals a rich large deviation structure of the free energy of logREMs with a deterministic background log potential. Many results are verified in the integrable circular logREM, by a replica-Coulomb gas integral approach. The related problem of common length (overlap) distribution is also considered. We provide a traveling-wave equation derivation of the LFT predictions announced in a precedent work.

  • 1. LPTENS - Laboratoire de Physique Théorique de l'ENS
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques