Discrete Derivative Asymptotics of the $\beta$-Hermite Eigenvalues

TL;DR

This paper studies the asymptotic behavior of the difference between empirical measures of the $eta$-Hermite matrix and its minor, revealing a deterministic limit linked to the Vershik-Kerov-Logan-Shepp curve and Gaussian fluctuations related to the Gaussian free field.

Contribution

It establishes the deterministic limit and Gaussian fluctuation results for the measure difference, connecting random matrix theory with geometric and field-theoretic concepts.

Findings

Deterministic limit identified with Vershik-Kerov-Logan-Shepp curve

Gaussian fluctuations correspond to a sectional derivative of the Gaussian free field

Asymptotic behavior characterized for $eta$-Hermite eigenvalues

Abstract

We consider the asymptotics of the difference between the empirical measures of the $\beta$-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik-Kerov-Logan-Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.