An exact mapping is established between the $c geq 25$ Liouville field theory (LFT) and the Gibbs measure statistics of a thermal particle in a 2D Gaussian Free Field plus a logarithmic confining potential. The probability distribution of the position of the minimum of the energy landscape is obtained exactly by combining the conformal bootstrap and one-step replica symmetry breaking methods. Operator product expansions in LFT allow to unveil novel universal behaviours of the “log-correlated Random Energy Models'' class. Applications will include multi-fractality (inverse participation ratios and their corrections) and the overlap distribution in the directed polymer on a Cayley tree model. Ref. https://arxiv.org/abs/1611.02193, collaborators: A. Rosso, R. Santachiara (LPTMS), P. Le Doussal (LPTENS),

LPTMS