An estimate for $β$-Hermite ensembles via the zeros of Hermite polynomials

TL;DR

This paper studies how close the eigenvalue vector of a $\beta$-Hermite ensemble is to the vector of ordered Hermite polynomial zeros in the large-\(\beta\) regime. It leverages a freezing central limit theorem to obtain an explicit limiting covariance with eigenstructure and then derives quantitative tail bounds for the distance between the scaled random vector and the deterministic zero vector, including a concrete bound with decay terms. The main contributions are an explicit tail bound for $P\left(\|X_{k,N}/\sqrt{2k}-z_N\|_2>\varepsilon\right)$, identification of the covariance eigenvalues $1,\dots,N$, and a comparison to Dette-Imhof (2009) that clarifies the bound’s informativeness across regimes $k$ vs. $N$. This advances understanding of fluctuations in high-\(\beta\) Hermite ensembles and provides practical quantitative estimates for convergence to the Hermite zeros.

Abstract

Let $X$ be an $N$-dimensional random vector which describes the ordered eigenvalues of a $β$-Hermite ensemble, and let $z$ the vector containing the ordered zeros of the Hermite poynomial $H_N$. We present an explicit estimate for $P(\|X-z\|_2\geε)$ for small $ε>0$ and large parameters $β$. The proof is based on a central limit theorem for these ensembles for $β\to\infty$ with explicit eigenvalues of the covariance matrices of the limit. The estimate is similar to previous estimates of Dette and Imhof (2009).

Theorems & Definitions (9)

• Lemma 2.1
• proof
• Theorem 2.2
• Corollary 2.3
• proof
• Lemma 2.4
• Proposition 2.5
• Corollary 2.6
• proof : Proof of Proposition \ref{['main-result-a']}