Glassy phase and freezing of log-correlated Gaussian potentials

In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.