Boris Hanin (MIT) Airy Scaling of Random Hermite Functions at the Caustic

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Random Hermite functions are eigenfunctions of fixed energy for the isotropic harmonic oscillator in R^d. They therefore transition from highly oscillatory to exponentially damped at the caustic, the analog of the edge of the spectrum in random matrix theory. The purpose of this talk is to explain how the Airy kernel and its relatives appear in the scaling limit of random Hermite functions around the caustic as Planck's constant h goes to 0. Rather than being the kernel of the determinantal process, the Airy kernel will appear as the covariance function of a limiting Gaussian field.

This is joint work with Steve Zelditch and Peng Zhou.