Characteristic polynomials of random matrices at edge singularities

We have discussed earlier the correlation functions of the random variables det(lambda-X) in which X is a random matrix. In particular, the moments of the distribution of these random variables are universal functions, when m easured in the appropriate units of the level spacing. When the lambda's, i nstead of belonging to the bulk of the spectrum, approach the edge, a cross over takes place to an Airy or to a Bessel problem, and we consider here th ese modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various different phe nomenona may occur and one can tune the spectrum of this source matrix to o ther critical points. Again there are remarkably simple formulas for arbitr ary source matrices, which allow us to compute the moments of the character istic polynomials in these cases as well.