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Biorthogonal ensembles of derivative type

Tom Claeys, Jiyuan Zhang

TL;DR

The paper develops a unified framework for biorthogonal ensembles of derivative type, proving a general double contour integral expression for the correlation kernel and showing that the derivative structure is both necessary and sufficient for such a representation. It then establishes two new hard-edge limit kernels: a deformed Bessel-type kernel arising from additive LUE perturbations and a family of Muttalib-Borodin deformations yielding Wright-like generalized Bessel kernels, with precise contour forms and conditions on the weight via PF and related spaces. The results are underpinned by a functional-analytic setup connecting w and W, a closed-form partition function, and confluent limits, and are complemented by a detailed asymptotic analysis that highlights universality at hard edges and across derivative-type perturbations. Collectively, this work broadens the catalog of solvable biorthogonal ensembles and provides explicit kernels for new universality classes in random matrix theory and related probabilistic models.

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.

Biorthogonal ensembles of derivative type

TL;DR

The paper develops a unified framework for biorthogonal ensembles of derivative type, proving a general double contour integral expression for the correlation kernel and showing that the derivative structure is both necessary and sufficient for such a representation. It then establishes two new hard-edge limit kernels: a deformed Bessel-type kernel arising from additive LUE perturbations and a family of Muttalib-Borodin deformations yielding Wright-like generalized Bessel kernels, with precise contour forms and conditions on the weight via PF and related spaces. The results are underpinned by a functional-analytic setup connecting w and W, a closed-form partition function, and confluent limits, and are complemented by a detailed asymptotic analysis that highlights universality at hard edges and across derivative-type perturbations. Collectively, this work broadens the catalog of solvable biorthogonal ensembles and provides explicit kernels for new universality classes in random matrix theory and related probabilistic models.

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.
Paper Structure (18 sections, 9 theorems, 147 equations, 3 figures)

This paper contains 18 sections, 9 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Double contour integral expression for the correlation kernel
  4. Limit kernels for additive perturbations of LUE
  5. Limit kernels for Muttalib-Borodin deformations
  6. Outline
  7. Properties of biorthogonal ensembles of derivative type
  8. Function spaces for TEXT and TEXT
  9. Partition function
  10. Correlation kernel
  11. The confluent case
  12. Proof of Theorem \ref{['thm:algebraic']}
  13. Additive convolutions
  14. Pólya frequency functions
  15. Asymptotics of biorthogonal ensemble of derivative type
  16. ...and 3 more sections

Key Result

Theorem 1

Let $c_-<0<c_+$ and $N\in\mathbb N$.

Figures (3)

  • Figure 1: The vertical strip $c_-<\mathrm{Re}\, z<c_+$, the points $a_1,\ldots, a_N$, and the integration contours $\Sigma_N$ and $c+i\mathbb R$ in \ref{['kernel0']}.
  • Figure 2: The vertical strip $c_-<\mathrm{Re}\, z<c_+$ and the integration contours $\Sigma$ and $c+i\mathbb R$ in \ref{['KLUE+0']}.
  • Figure 3: The half-plane $\mathrm{Re}\, z>c_-$ and the integration contours $\Sigma$ and $c+i\mathbb R$ in \ref{['def:KMB']}.

Theorems & Definitions (40)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Theorem 1
  • Remark 1
  • Remark 2
  • ...and 30 more