Non-equilibrium Dynamics and Random Matrices

The quantum random energy model is a random matrix of Schroedinger type: a Laplacian on the hypercube plus a random potential. It features in various contexts from mathematical biology to quantum information theory as well as an effective description of (de)localization in many particle systems. Among its interesting properties is a first-order phase transition of the ground-state depending on the strength of the disorder. In this talk, I will focus on the description of the nature of the low-energy spectrum of this model. I will also mention some conjectures on the occurence of extended eigenstates due to resonant delocalization in this model. The latter is mechanism which allows the formation of extended eigenstates which nevertheless violate quantum ergodicity. It can be proven to occur in an oversimplification of the model: the random Schroedinger operator on a complete graph.