Torben Krueger: Spectral Universality for Random Matrices: From the Global to the Local Scale

The spectral statistics of large dimensional self-adjoint random matrices often exhibits universal behavior. On the global spectral scale the density of states depends only on the first two moments of the matrix entries and follows a universal shape at all its singular points, i.e. whenever it vanishes. On the local scale the joint distribution of a finite number of eigenvalues depends only on the symmetry type of the random matrix (Wigner-Dyson-Mehta spectral universality). We present recent results and methods that establish such spectral universality properties from the global down to the smallest spectral scale for a wide range of random matrix models, including matrices with general expectation and correlated entries. [Joint work with the Erdős group at IST Austria]