2010 AverageCharacteristicPolynomial

(Delvaux, 2010) ⇒ Steven Delvaux. (2010). “Average Characteristic Polynomials for Multiple Orthogonal Polynomial Ensembles.” In: Journal of Approximation Theory, 162(5). doi:10.1016/j.jat.2009.11.008

Subject Headings: Determinantal Point Process.

Notes

Cited By

Quotes

Author Keywords

(Block) Hankel determinant; Average characteristic polynomial; Christoffel-Darboux kernel; Determinantal point process; Multiple/matrix orthogonal polynomials; Riemann-Hilbert problem; Schur complement

Abstract

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann-Hilbert matrix <matrix>Y</matrix> consisting of four blocks [math]\displaystyle{ Y_{1,1} }[/math], [math]\displaystyle{ Y_{1,2} }[/math], [math]\displaystyle{ Y_{2,1} }[/math] and [math]\displaystyle{ Y_{2,2} }[/math]. In this paper, we show that detY"1","1 (detY"2","2) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize the classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel-Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.

References

,

	Author	volume	Date Value	title	type	journal	titleUrl	doi	note	year
2010 AverageCharacteristicPolynomial	Steven Delvaux			Average Characteristic Polynomials for Multiple Orthogonal Polynomial Ensembles				10.1016/j.jat.2009.11.008		2010