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

Gopal Goel; Andrew Ahn

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arXiv · 1809.06804

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

Gopal Goel·Andrew Ahn

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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.

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Gopal Goel, Andrew Ahn. 2018-09-18. Discrete Derivative Asymptotics of the $\beta$-Hermite Eigenvalues. https://arxiv.org/abs/1809.06804

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