Rigidity of the Stochastic Airy Operator

Abstract

Abstract We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres [22] 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 [9]. 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 in [24].