Rigidity of the Stochastic Airy Operator

TL;DR

This paper establishes the rigidity of the spectrum of the stochastic Airy operator, confirming the stability of the Airy-$eta$ point process and the soft-edge limits of Gaussian $eta$-Ensembles for all $eta>0$, solving a notable open problem.

Contribution

It proves the spectral rigidity of the stochastic Airy operator for various boundary conditions, extending the understanding of point process stability and solving an open problem in the field.

Findings

Spectral rigidity of the stochastic Airy operator established.

Rigidity of the Airy-$eta$ point process confirmed.

Soft-edge limits of Gaussian $eta$-Ensembles shown to be rigid.

Abstract

We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789--1858, 2017) for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-$\beta$ point process and the soft-edge limit of rank-$1$ perturbations of Gaussian $\beta$-Ensembles for any $\beta>0$, and solves an open problem mentioned in a previous work of Bufetov, Nikitin, and Qiu (Mosc. Math. J., 19(2):217--274, 2019). Our proof uses a combination of the semigroup theory of the stochastic Airy operator and the techniques for studying insertion and deletion tolerance of point processes developed by Holroyd and Soo (Electron. J. Probab., 18:no. 74, 24, 2013).