Characteristic polynomials of random matrices at edge singularities

TL;DR

This paper explores the universal behavior of characteristic polynomials of random matrices at spectral edges, including effects of external sources, revealing new phenomena and simple formulas for moments.

Contribution

It introduces new universality classes at spectral edges and provides explicit formulas for moments with external matrix sources.

Findings

Universal correlation functions at spectral edges

New phenomena with external matrix sources

Explicit formulas for moments of characteristic polynomials

Abstract

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.