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Improved concentration of Laguerre and Jacobi ensembles

Yichen Huang, Aram W. Harrow

Abstract

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.

Improved concentration of Laguerre and Jacobi ensembles

Abstract

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.
Paper Structure (13 sections, 15 theorems, 88 equations)

This paper contains 13 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Laguerre ensemble
  4. Jacobi ensemble
  5. Pointwise approximation
  6. Moments
  7. Proofs
  8. Laguerre ensemble: Proofs of Theorem \ref{['thm:l3']} and Fact \ref{['fact:M1L']}
  9. Jacobi ensemble
  10. Pointwise approximation: Proof of Theorem \ref{['thm:j2']}
  11. Moments: Proof of Theorem \ref{['thm:moment']}
  12. Proof of Eq. (\ref{['eq:app']})
  13. Moments of the Hermite ensemble

Key Result

Theorem 1

For any $0<\epsilon<1$,

Theorems & Definitions (28)

  • Definition 1: Laguerre ensemble
  • Theorem 1: Theorem 2.1 in Ref. DI07
  • Corollary 1
  • Theorem 2: Theorem 2.4 in Ref. DI07
  • Theorem 3
  • Corollary 2
  • Definition 2: Jacobi ensemble
  • Theorem 4: Theorem 2.1 in Ref. DN09
  • Corollary 3
  • Theorem 5
  • ...and 18 more