Derivation of determinantal structures for random matrix ensembles in a new way

Abstract

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can coin our approach "supersymmetry without supersymmetry" because we use structures from supersymmetry without actually mapping onto superspaces. We address two kinds of integrals which cover a wide range of applications for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.

36 pages; 2 tables

References in corpus (9)

• Superbosonization of invariant random matrix ensembles
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• Random matrix analysis of the QCD sign problem for general topology
• Arbitrary rotation invariant random matrix ensembles and supersymmetry: orthogonal and unitary-symplectic case
• The supersymmetry method of random matrix theory
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Cited by in corpus (36)

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• Exact Relation between Singular Value and Eigenvalue Statistics
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• A new approach to derive Pfaffian structures for random matrix ensembles
• Spectral Properties of the Wilson Dirac Operator and random matrix theory
• Secrecy Rate of Cooperative MIMO in Presence of a Location Constrained Eavesdropper
• Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• Harmonic analysis for rank-1 Randomised Horn Problems
• GUE-chGUE Transition preserving Chirality at finite Matrix Size
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Additive Matrix Convolutions of Pólya Ensembles and Polynomial Ensembles
• Spectral statistics for the difference of two Wishart matrices
• Average characteristic polynomials in the two-matrix model
• On the Efetov-Wegner terms by diagonalizing a Hermitian supermatrix
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence
• Products of Complex Rectangular and Hermitian Random Matrices
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Optical MIMO communication with unequal power allocation to channels
• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case
• Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems
• Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics
• Duality and integrability of supermatrix model with external source
• Average mutual information for random fermionic Gaussian quantum states
• Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case
• Fox H-kernel and $\theta$-deformation of the Cauchy two-matrix model and Bures ensemble
• Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble
• Correlation functions between singular values and eigenvalues