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Stability of Mesoscopic Fluctuations of Orthogonal Polynomial Ensembles Under Sparse Decaying Perturbations

Daniel Ofner

Abstract

We study the stability of the mesoscopic fluctuations of certain orthogonal polynomial ensembles on the real line utilizing the recurrence relation of the associated orthogonal polynomials. We prove that under a sparse enough decaying perturbation of the recurrence coefficients the limiting distribution is stable. As a corollary we prove a mesoscopic central limit theorem (at any scale) for a family of singular continuous measures on $[-2,2]$.

Stability of Mesoscopic Fluctuations of Orthogonal Polynomial Ensembles Under Sparse Decaying Perturbations

Abstract

We study the stability of the mesoscopic fluctuations of certain orthogonal polynomial ensembles on the real line utilizing the recurrence relation of the associated orthogonal polynomials. We prove that under a sparse enough decaying perturbation of the recurrence coefficients the limiting distribution is stable. As a corollary we prove a mesoscopic central limit theorem (at any scale) for a family of singular continuous measures on .

Paper Structure

This paper contains 10 sections, 9 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fredholm determinant identity
  4. Cumulant generating function and cumulants in terms of $J$.
  5. Comparison principle of cumulants.
  6. Mesoscopic stability- proof of Theorem \ref{['thm:main']}
  7. Step 0: Mesoscopic fluctuations in terms of Jacobi matrices
  8. Step 1: Decoupling.
  9. Step 2: Entry-wise estimations
  10. Step 3: Estimating the trace norm of a Hankel matrix

Key Result

Theorem 1.1

Let $J_0$ be the free Jacobi matrix. Then for every $0<\gamma<1,\:x_0\in \left(-2,2\right)$ and $\forall\beta\in (\gamma,1)$, if $J$ is as in eq:J=J0+V and $\mu$ is the associated spectral measure of $J$ and $e_1$, then for any $f\in C^1_c\left({\mathbb{R}}\right)$ and $m\in{\mathbb{N}}$ we have where ${\mathbb{E}}_{\mu_{0}},{\mathbb{E}}_{\mu}$ denote the expectations with respect to $\text{OPE}(

Theorems & Definitions (22)

  • Definition 1
  • Example 1
  • Theorem 1.1
  • Remark 1
  • Corollary 1
  • Theorem 1.2
  • proof
  • Proposition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • ...and 12 more