On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

Abstract

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian Orthogonal Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large-$N$ limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.

arXiv comment25 pages (2 figures); published version (conclusion added, minor changes)

References in corpus (30)

• Random Matrix Theories in Quantum Physics: Common Concepts
• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption
• GENERALIZED CIRCULAR ENSEMBLE OF SCATTERING MATRICES FOR A CHAOTIC CAVITY WITH NON-IDEAL LEADS
• Universal Impedance Fluctuations in Wave Chaotic Systems
• Characteristic polynomials of random matrices
• How many entries of a typical orthogonal matrix can be approximated by independent normals?
• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation
• Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles
• Quantum Chaotic Scattering in Microwave Resonators
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• An exact formula for general spectral correlation function of random Hermitian matrices
• Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach
• Duality in random matrix ensembles for all Beta
• Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems
• Derivation of determinantal structures for random matrix ensembles in a new way
• Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering
• Universal statistics of the local Green's function in quantum chaotic systems with absorption
• Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
• A new approach to derive Pfaffian structures for random matrix ensembles
• Singularity dominated strong fluctuations for some random matrix averages
• Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results
• Random-Matrix Theory of Quantum Size Effects on Nuclear Magnetic Resonance in Metal Particles
• Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Distribution of the Absorption by Chaotic States in Quantum Dots
• On Universality for Orthogonal Ensembles of Random Matrices
• On the ubiquity of the Cauchy distribution in spectral problems
• Universal K-matrix distribution in beta=2 Ensembles of Random Matrices

Cited by in corpus (15)

• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Fidelity susceptibility in Gaussian Random Ensembles
• Random matrix theory of quantum transport in chaotic cavities with non-ideal leads
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Resonance width distribution in RMT: Weak coupling regime beyond Porter-Thomas
• Experimental investigation of distributions of the off-diagonal elements of the scattering and the Wigner's $\hat{K}$ matrices for networks with broken time reversal invariance
• The Loss Surfaces of Neural Networks with General Activation Functions
• Distribution of Off-Diagonal Cross Sections in Quantum Chaotic Scattering: Exact Results and Data Comparison
• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Quantum chaos and level dynamics
• Statistics of off-diagonal entries of Wigner $K$-matrix for chaotic wave systems with absorption
• Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small $N$
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles
• Resistance distance distribution in large sparse random graphs