Biorthogonal ensembles of derivative type

TL;DR

This paper derives explicit correlation kernels for biorthogonal ensembles with derivative structures, enabling asymptotic analysis and revealing new limit kernels related to random matrix eigenvalue distributions.

Contribution

It introduces explicit double contour integral correlation kernels for a class of biorthogonal ensembles, expanding understanding of their asymptotic behaviors and limit kernels.

Findings

Derived explicit correlation kernels for biorthogonal ensembles.

Identified new classes of limit kernels related to random matrix models.

Showed the applicability of kernels to asymptotic analysis of eigenvalue distributions.

Abstract

In this paper, we prove that biorthogonal ensembles on the real line with a specific derivative structure admit an explicit correlation kernel of double contour integral form. We will demonstrate that this expression is a valuable starting point for asymptotic analysis and that our class of biorthogonal ensembles admits a large variety of limit kernels, by proving that two new classes of limit kernels can occur. The first type is a deformation of the hard edge Bessel kernel which arises in polynomial ensembles describing the eigenvalues of the sum of two random matrices, while the second type arises for Muttalib-Borodin type deformations of polynomial ensembles.