Abstract
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.
Highlights
We study correlation functions of characteristic polynomials in a sub-class of determinantal random point processes
Polynomial ensembles are characterised by the fact that one of the two determinants in the joint density of points is given by a Vandermonde determinant, while the other one is kept general
Polynomial ensembles appear in various contexts as the joint distribution of eigenvalues of random matrices, see [3,14,19,20,27]
Summary
We study correlation functions of characteristic polynomials in a sub-class of determinantal random point processes. Most recently the study of eigenvector statistics of random matrices has seen a revival, and in this context expectation values of ratios of characteristic polynomials in polynomial ensembles arise [23,24] This has been one of the starting points of the present work. Theorem 2.2 says that any polynomial ensemble is a Giambelli compatible point process in the sense of Borodin, Olshanski, and Strahov [12] This leads to Theorem 2.3, expressing the expectation value of the ratio of an equal number of characteristic polynomials as a determinant of a single ratio, generalising [7, Theorem 3.3] to polynomial ensembles. Appendix A collects properties of the Vandermonde determinant, when adding or removing factors
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