Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Abstract

We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta =2$ symmetry class.

34pages

References in corpus (18)

• Level-Spacing Distributions and the Airy Kernel
• Fredholm Determinants, Differential Equations and Matrix Models
• Level-Spacing Distributions and the Bessel Kernel
• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Random Matrices close to Hermitian or unitary: overview of methods and results
• Characteristic polynomials of random matrices
• Systematic Analytical Approach to Correlation Functions of Resonances in Quantum Chaotic Scattering
• Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation
• An exact formula for general spectral correlation function of random Hermitian matrices
• Replica Limit of the Toda Lattice Equation
• Replica field theories, Painleve transcendents, and exact correlation functions
• Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
• On correlation functions of characteristic polynomials for chiral Gaussian Unitary Ensemble
• Microscopic Spectrum of the QCD Dirac Operator in Three Dimensions
• Characteristic polynomials of random matrices at edge singularities
• Character Expansions for the Orthogonal and Symplectic Groups
• Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations
• Det-Det Correlations for Quantum Maps: Dual Pair and Saddle-Point Analyses

Cited by in corpus (38)

• Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• On the averages of characteristic polynomials from classical groups
• Applications and generalizations of Fisher-Hartwig asymptotics
• Relating the Bures measure to the Cauchy two-matrix model
• Fidelity susceptibility in Gaussian Random Ensembles
• FZZT branes in JT gravity and topological gravity
• On the distribution of maximum value of the characteristic polynomial of GUE random matrices
• Random matrix analysis of the QCD sign problem for general topology
• Ratios of characteristic polynomials in complex matrix models
• Universal shocks in random matrix theory
• On matrix model partition functions for QCD with chemical potential
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials
• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• Hilbert-space geometry of random-matrix eigenstates
• On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
• On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Universal K-matrix distribution in beta=2 Ensembles of Random Matrices
• Statistics of off-diagonal entries of Wigner $K$-matrix for chaotic wave systems with absorption
• On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices
• The n-level spectral correlations for chaotic systems
• Average characteristic polynomials in the two-matrix model
• Determinantal point processes and fermion quasifree states
• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• On the moments of characteristic polynomials
• Characteristic polynomials for random band matrices near the threshold
• Random matrix approach to three-dimensional QCD with a Chern-Simons term
• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
• Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles
• Survival Probability of a Doorway State in regular and chaotic environments
• Symmetric Function Theory and Unitary Invariant Ensembles
• Duality and integrability of supermatrix model with external source
• Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality
• Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble