Hard edge asymptotics of correlation functions between singular values and eigenvalues

Abstract

Any square complex matrix of size $n\times n$ can be partially characterized by its $n$ eigenvalues and/or $n$ singular values. While no one-to-one correspondence exists between those two kinds of values on a deterministic level, for random complex matrices drawn from a bi-unitarily invariant ensemble, a bijection exists between the underlying singular value ensemble and the corresponding eigenvalue ensemble. This enabled the recent finding of an explicit formula for the joint probability density between $1$ eigenvalue and $k$ singular values, coined $1,k$-point function. We derive here the large $n$ asymptotic of the $1,k$-point function around the origin (hard edge) for a large subclass of bi-unitarily invariant ensembles called polynomial ensembles and its subclass Pólya ensembles. This latter subclass contains all Meijer-G ensembles and, in particular, Muttalib-Borodin ensembles and the classical Wishart-Laguerre (complex Ginibre), Jacobi (truncated unitary), Cauchy-Lorentz ensembles. We show that the latter three ensembles share the same asymptotic of the $1,k$-point function around the origin. In the case of Jacobi ensembles, there exists another hard edge for the singular values, namely the upper edge of their support, which corresponds to a soft edge for the eigenvalue (soft-hard edge). We give the explicit large $n$ asymptotic of the $1,k$-point function around this soft-hard edge.

arXiv commentResults remain unchanged. This version includes minor revisions, corrections, and additional comments compared to the published version. 44 pages, 1 figure, 1 appendix

References in corpus (24)

• Random-Matrix Theory of Quantum Transport
• Biorthogonal Quantum Mechanics
• Biorthogonal ensembles
• Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition
• Topological Amplification in Photonic Lattices
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Singular value statistics of matrix products with truncated unitary matrices
• Singular values of the Dirac operator in dense QCD-like theories
• Universal microscopic correlation functions for products of truncated unitary matrices
• Energy correlations for a random matrix model of disordered bosons
• Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification
• Products of Random Matrices from Polynomial Ensembles
• Exact Relation between Singular Value and Eigenvalue Statistics
• Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition
• Scaling in the eigenvalue fluctuations of the empirical correlation matrices
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Krylov space approach to Singular Value Decomposition in non-Hermitian systems
• Polynomial Ensembles and Pólya Frequency Functions
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Singular Value Decomposition and Its Blind Spot for Quantum Chaos in Non-Hermitian Sachdev-Ye-Kitaev Models
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence
• Winding Number Statistics of a Parametric Chiral Unitary Random Matrix Ensemble
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case

Cited by in corpus (1)

• Correlation functions between singular values and eigenvalues