In the study of disordered quantum systems, it is believed that a strong dichotomy should occur between two phases: a delocalized (or conducting) phase and a localized (or insulating) phase. While this is far from being proved in all generality, the study of large symmetric random matrices, which model simple systems, allows us to describe in a precise way the delocalized phase. In this talk, we will present several delocalization estimates on eigenvectors of random matrices such as sharp upper bounds on their infinite norm, asymptotic distribution of entries, and quantum unique ergodicity or quantum weak mixing properties. We will also present ideas of proof based on a dynamical method on symmetric random matrices.

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