Hard edge asymptotics of correlation functions between singular values and one eigenvalue

Abstract Any square complex matrix of size n × 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> -point function. We derive here the large n asymptotic of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> -point function around this soft-hard edge.