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).
