  An exact mapping is established between the $c\geq25$ 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 class. High precision numerical
tests are given.
