Work

Correlation functions between singular values and eigenvalues

Year: 2025

Type: article

Abstract: Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure onjeigenvalues andksingular values that we coinj,k-point correlation measure. We fully derive allj,k-point correlation measures in the simplest cases for matrices of sizen= 1andn= 2 . Forn> 2, we find a general formula for the1, 1-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. MSC Classification: 60B20 , 15B52 , 43A90 , 42B10 , 42C05

Source: Random Matrices Theory and Application

DOI: 10.1142/s2010326325500248

Authors: Matthias Allard, Mario Kieburg

Institutions: The University of Melbourne, KU Leuven

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: KU Leuven, Australian Research Council, Faculty of Science, University of Melbourne

Award: Expanding and linking random matrix theory