The cubic law, the invariance principle, and related topics in the theory of analytic functions of random matrices (Q2722275)

The important question in the well-developed theory of random Gram matrices is how to describe the limit normalized spectral functions (n.s.f.) of an analytic function of random matrices. One of them is, for example, how to find the limit of the n.s.f. of the so-called double Gram matrices which are constructed by random matrices. Some nonlinear equations for the Stieltjes transform of the n.s.f. of this matrix are derived and relations of this assertion with simple results for the maximal eigenvalue of the matrix are established in this paper. In particular, the so-called cubic law is proved. Invariance principle and other related topics are investigated.