Glassy phase and freezing of log-correlated Gaussian potentials

Understand

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.