Characteristic polynomials of random matrices at edge singularities

We have discussed earlier the correlation functions of the random variables $\det \left(\lambda -X\right)$ in which X is a random matrix. In particular, the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the $\lambda ’\mathrm{s},$ instead of belonging to the bulk of the spectrum, approach the edge, a crossover takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various different phenomenona may occur and one can tune the spectrum of this source matrix to other critical points. Again there are remarkably simple formulas for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.