Rational functions of Wigner matrices and scattering in quantum dots

We consider general self-adjoint rational functions in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish for these models the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. Moreover, in the framework of the developed theory, we study the density of transmission eigenvalues in the random matrix model for transport in quantum dots coupled to a chaotic environment.

This is a joint work with Laszlo Erdös and Torben Krüger.