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

TL;DR

This paper provides an explicit probabilistic estimate for the deviation between eigenvalues of a $eta$-Hermite ensemble and the zeros of Hermite polynomials, especially for large $eta$, using a CLT approach.

Contribution

It introduces a new explicit estimate for the probability that the eigenvalues deviate from polynomial zeros, based on a CLT for large $eta$ in $eta$-Hermite ensembles.

Findings

Derived an explicit estimate for eigenvalue deviations

Utilized a CLT with explicit covariance eigenvalues

Results extend previous estimates by Dette and Imhof (2009)

Abstract

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