Characteristic polynomials of random matrices at edge singularities

Details Characteristic polynomials of random matrices at edge singularities Characteristic polynomials of random matrices at edge singularities

2000-04-13

arxiv.org/abs/math-ph/0004018v1

Physics

Mathematical Physics

22 pages, latex

Scientific paper

10.1103/PhysRevE.62.3558

We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ 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 $\la$'s, instead 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 these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of $X$, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.

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