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Phillips M. J. - One of the best experts on this subject based on the ideXlab platform.

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    'IOP Publishing', 2010
    Co-Authors: Akemann Gernot, Kieburg M., Phillips M. J.
    Abstract:

    Akemann G, Kieburg M, Phillips MJ. Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices. J. Phys. A. 2010;43(37): 375207.We apply the method of skew-orthogonal polynomials (SOP) in the Complex planeto asymmetric random matrices with real elements, belonging to two differentclasses. Explicit integral representations valid for arbitrary weight functionsare derived for the SOP and for their Cauchy transforms, given as expectationvalues of traces and determinants or their inverses, respectively. Our proofuses the fact that the joint probability distribution function for allcombinations of real Eigenvalues and Complex Conjugate Eigenvalue pairs can bewritten as a product. Examples for the SOP are given in terms of Laguerrepolynomials for the chiral ensemble (also called the non-Hermitian realWishart-Laguerre ensemble), both without and with the insertion ofcharacteristic polynomials. Such characteristic polynomials play the role ofmass terms in applications to Complex Dirac spectra in field theory. Inaddition, for the elliptic real Ginibre ensemble we recover the SOP ofForrester and Nagao in terms of Hermite polynomials

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    'IOP Publishing', 2010
    Co-Authors: Akemann G., Kieburg M., Phillips M. J.
    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.Comment: 27 pages; v2: typos corrected and references adde

  • The chiral Gaussian two-matrix ensemble of real asymmetric matrices
    'IOP Publishing', 2009
    Co-Authors: Akemann Gernot, Phillips M. J., Sommers H. -j.
    Abstract:

    Akemann G, Phillips MJ, Sommers H-J. The chiral Gaussian two-matrix ensemble of real asymmetric matrices. J.Phys.A. 2009;43(8):085211.We solve a family of Gaussian two-matrix models with rectangular Nx(N+v)matrices, having real asymmetric matrix elements and depending on anon-Hermiticity parameter mu. Our model can be thought of as the chiralextension of the real Ginibre ensemble, relevant for Dirac operators in thesame symmetry class. It has the property that its Eigenvalues are either real,purely imaginary, or come in Complex Conjugate Eigenvalue pairs. The Eigenvaluejoint probability distribution for our model is explicitly computed, leading toa non-Gaussian distribution including K-Bessel functions. All n-point densitycorrelation functions are expressed for finite N in terms of a Pfaffian form.This contains a kernel involving Laguerre polynomials in the Complex plane as abuilding block which was previously computed by the authors. This kernel can beexpressed in terms of the kernel for Complex non-Hermitian matrices,generalising the known relation among ensembles of Hermitian random matrices.Compact expressions are given for the density at finite N as an example, aswell as its microscopic large-N limits at the origin for fixed v at strong andweak non-Hermiticity

  • The chiral Gaussian two-matrix ensemble of real asymmetric matrices
    'IOP Publishing', 2009
    Co-Authors: Akemann G., Phillips M. J., Sommers H. -j.
    Abstract:

    We solve a family of Gaussian two-matrix models with rectangular Nx(N+v) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter mu. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its Eigenvalues are either real, purely imaginary, or come in Complex Conjugate Eigenvalue pairs. The Eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the Complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for Complex non-Hermitian matrices, generalising the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed v at strong and weak non-Hermiticity.Comment: 31 pages, 6 figure

Akemann Gernot - One of the best experts on this subject based on the ideXlab platform.

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    'IOP Publishing', 2010
    Co-Authors: Akemann Gernot, Kieburg M., Phillips M. J.
    Abstract:

    Akemann G, Kieburg M, Phillips MJ. Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices. J. Phys. A. 2010;43(37): 375207.We apply the method of skew-orthogonal polynomials (SOP) in the Complex planeto asymmetric random matrices with real elements, belonging to two differentclasses. Explicit integral representations valid for arbitrary weight functionsare derived for the SOP and for their Cauchy transforms, given as expectationvalues of traces and determinants or their inverses, respectively. Our proofuses the fact that the joint probability distribution function for allcombinations of real Eigenvalues and Complex Conjugate Eigenvalue pairs can bewritten as a product. Examples for the SOP are given in terms of Laguerrepolynomials for the chiral ensemble (also called the non-Hermitian realWishart-Laguerre ensemble), both without and with the insertion ofcharacteristic polynomials. Such characteristic polynomials play the role ofmass terms in applications to Complex Dirac spectra in field theory. Inaddition, for the elliptic real Ginibre ensemble we recover the SOP ofForrester and Nagao in terms of Hermite polynomials

  • The chiral Gaussian two-matrix ensemble of real asymmetric matrices
    'IOP Publishing', 2009
    Co-Authors: Akemann Gernot, Phillips M. J., Sommers H. -j.
    Abstract:

    Akemann G, Phillips MJ, Sommers H-J. The chiral Gaussian two-matrix ensemble of real asymmetric matrices. J.Phys.A. 2009;43(8):085211.We solve a family of Gaussian two-matrix models with rectangular Nx(N+v)matrices, having real asymmetric matrix elements and depending on anon-Hermiticity parameter mu. Our model can be thought of as the chiralextension of the real Ginibre ensemble, relevant for Dirac operators in thesame symmetry class. It has the property that its Eigenvalues are either real,purely imaginary, or come in Complex Conjugate Eigenvalue pairs. The Eigenvaluejoint probability distribution for our model is explicitly computed, leading toa non-Gaussian distribution including K-Bessel functions. All n-point densitycorrelation functions are expressed for finite N in terms of a Pfaffian form.This contains a kernel involving Laguerre polynomials in the Complex plane as abuilding block which was previously computed by the authors. This kernel can beexpressed in terms of the kernel for Complex non-Hermitian matrices,generalising the known relation among ensembles of Hermitian random matrices.Compact expressions are given for the density at finite N as an example, aswell as its microscopic large-N limits at the origin for fixed v at strong andweak non-Hermiticity

M. J. Phillips - One of the best experts on this subject based on the ideXlab platform.

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    2014
    Co-Authors: G. Akemann, M. Kieburg, M. J. Phillips
    Abstract:

    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

  • the chiral gaussian two matrix ensemble of real asymmetric matrices
    Journal of Physics A, 2010
    Co-Authors: G. Akemann, M. J. Phillips, Hansjurgen Sommers
    Abstract:

    We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its Eigenvalues are either real, purely imaginary or come in Complex Conjugate Eigenvalue pairs. The Eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the Complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for Complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed ν at strong and weak non-Hermiticity.

G. Akemann - One of the best experts on this subject based on the ideXlab platform.

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    2014
    Co-Authors: G. Akemann, M. Kieburg, M. J. Phillips
    Abstract:

    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

  • the chiral gaussian two matrix ensemble of real asymmetric matrices
    Journal of Physics A, 2010
    Co-Authors: G. Akemann, M. J. Phillips, Hansjurgen Sommers
    Abstract:

    We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its Eigenvalues are either real, purely imaginary or come in Complex Conjugate Eigenvalue pairs. The Eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the Complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for Complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed ν at strong and weak non-Hermiticity.

Akemann G. - One of the best experts on this subject based on the ideXlab platform.

  • Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
    'IOP Publishing', 2010
    Co-Authors: Akemann G., Kieburg M., Phillips M. J.
    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.Comment: 27 pages; v2: typos corrected and references adde

  • The chiral Gaussian two-matrix ensemble of real asymmetric matrices
    'IOP Publishing', 2009
    Co-Authors: Akemann G., Phillips M. J., Sommers H. -j.
    Abstract:

    We solve a family of Gaussian two-matrix models with rectangular Nx(N+v) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter mu. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its Eigenvalues are either real, purely imaginary, or come in Complex Conjugate Eigenvalue pairs. The Eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the Complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for Complex non-Hermitian matrices, generalising the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed v at strong and weak non-Hermiticity.Comment: 31 pages, 6 figure