Supersymmetric approach to random band matrices

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition even in 1d. In this talk we will discuss an application of the supersymmetric method (SUSY) to the analysis of the bulk local regime of some specific types of RBM. We present rigorous SUSY results about the crossover for 1d RBM on the level of characteristic polynomials, as well as some progress in studying of the density of states and usual second correlation function.