  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.

Probability