Universality of "level curvature" distribution for large random matrices: systematic analytical approaches

We perform an extensive analytical study of distributions of "level curvatures'' (the second derivatives of eigenvalues with respect to a perturbation parameter) for different classes of random matrices. First, we consider the case of three Gaussian ensembles: GUE, GOE and GSE. This part of our calculation is complementary to that done recently by von Oppen [22, 23], but evaluation goes along different lines and allows to treat all the three cases uniformly. In the second part of the paper we exploit completely another method allowing to treat the problem analytically for the broad class of disordered systems subject to time-reversal symmetry breaking perturbation. That gives us a possibility to prove the conjecture by Zakrzewski and Delande [17] for the ensemble of symmetric sparse random matrices.