Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

TY - JOUR

T1 - Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

AU - Akemann, G.

AU - Kieburg, M.

AU - Philips, M. J.

PY - 2010/9/17

Y1 - 2010/9/17

N2 - 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 the 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.

AB - 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 the 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.

U2 - 10.1088/1751-8113/43/37/375207

DO - 10.1088/1751-8113/43/37/375207

M3 - Journal article

SN - 1751-8113

VL - 43

SP - 375207

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 37

ER -