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Title: Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
Authors: Akemann, G
Kieburg, M
Phillips, M J
Keywords: Skew-orthogonal Laguerre polynomials;Real asymmetric random matrices;Characteristic polynomials;Cauchy transform
Issue Date: 2010
Publisher: IoP
Citation: Journal of Physics A: Mathematical and Theoretical, 43 (2010) 375207
Abstract: We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.
ISSN: 1751-8113
Appears in Collections:Dept of Mathematics Research Papers
Mathematical Sciences

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