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Multiple orthogonal polynomial ensembles of derivative type

Thomas Wolfs

TL;DR

The paper develops a transformative framework to classify and construct polynomial ensembles of derivative type (MDT in the multiplicative setting via the Mellin transform and ADT in the additive setting via the Laplace transform). It proves a complete transform-based characterization: MDT corresponds to Gamma-product Mellin transforms and exponential-integral Laplace transforms, yielding explicit parameterizations and positivity conditions; JUE, LUE, and GUE emerge as canonical MDT/ADT examples. The work then connects these ensembles to finite free probability, showing how products and sums of random matrices preserve (or decompose into) polynomial ensembles under finite free convolution, and providing partial answers to recent open problems in Nijmegen. Collectively, the results supply a unified, transform-driven method to build and analyze multiple orthogonal polynomial ensembles with derivative-type structure and to understand their behavior under finite free convolution.

Abstract

We characterize the biorthogonal ensembles that are both a multiple orthogonal polynomial ensemble and a polynomial ensemble of derivative type (also called a Pólya ensemble). We focus on the two notions of derivative type that typically appear in connection with the squared singular values of products of invertible random matrices and the eigenvalues of sums of Hermitian random matrices. Essential in the characterization is the use of the Mellin and Laplace transform: we show that the derivative type structure, which is a priori analytic in nature, becomes algebraic after applying the appropriate transform. Afterwards, we explain how these notions of derivative type can be used to provide a partial solution to an open problem related to orthogonality of the finite finite free multiplicative and additive convolution from finite free probability. In particular, we obtain families of multiple orthogonal polynomials that (de)compose naturally using these convolutions.

Multiple orthogonal polynomial ensembles of derivative type

TL;DR

The paper develops a transformative framework to classify and construct polynomial ensembles of derivative type (MDT in the multiplicative setting via the Mellin transform and ADT in the additive setting via the Laplace transform). It proves a complete transform-based characterization: MDT corresponds to Gamma-product Mellin transforms and exponential-integral Laplace transforms, yielding explicit parameterizations and positivity conditions; JUE, LUE, and GUE emerge as canonical MDT/ADT examples. The work then connects these ensembles to finite free probability, showing how products and sums of random matrices preserve (or decompose into) polynomial ensembles under finite free convolution, and providing partial answers to recent open problems in Nijmegen. Collectively, the results supply a unified, transform-driven method to build and analyze multiple orthogonal polynomial ensembles with derivative-type structure and to understand their behavior under finite free convolution.

Abstract

We characterize the biorthogonal ensembles that are both a multiple orthogonal polynomial ensemble and a polynomial ensemble of derivative type (also called a Pólya ensemble). We focus on the two notions of derivative type that typically appear in connection with the squared singular values of products of invertible random matrices and the eigenvalues of sums of Hermitian random matrices. Essential in the characterization is the use of the Mellin and Laplace transform: we show that the derivative type structure, which is a priori analytic in nature, becomes algebraic after applying the appropriate transform. Afterwards, we explain how these notions of derivative type can be used to provide a partial solution to an open problem related to orthogonality of the finite finite free multiplicative and additive convolution from finite free probability. In particular, we obtain families of multiple orthogonal polynomials that (de)compose naturally using these convolutions.

Paper Structure

This paper contains 15 sections, 40 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Mellin transform
  5. Laplace transform
  6. Derivative type
  7. Characterization
  8. Multiplicative setting
  9. Additive setting
  10. Examples
  11. Multiplicative setting
  12. Additive setting
  13. Finite free probability
  14. Multiplicative setting
  15. Additive setting

Key Result

Theorem 2.1

Let $w_1,\dots,w_r\in L^1_{\mathcal{M},\Sigma}(\mathbb{R}_{>0})$, with $\mathbb{Z}_{\geq 1}\subset \Sigma$, be real-valued and a.e. positive. The set $\{x^{k-1}w_j(x) \mid k=1,\dots,n_j,j=1,\dots,r\}$ is of multiplicative derivative type w.r.t. $\omega_{\vec{n}}$ for all $\vec{n}\in\mathcal{S}^r$ if for some $a_i\in\mathbb{C}$, $b_i\in\mathbb{R}$, $c\in\mathbb{R}$ and $d_1,d_2:\{1,\dots,r\}\to\{0,

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • ...and 77 more