From random matrices to random analytic functions

Singular points of random matrix-valued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n× n matrices. Singular points are those (random) z where the matrix becomes singular, that is, the zeros of the determinant. This notion was introduced in the Ph.D thesis [10] of the author, where some basic facts were found. Of course, singular points are just the zeros of the (random analytic function) determinant, so in what sense is this concept novel? In case of random matrices as well as random analytic functions, the following features may be observed.