Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models

Let $\Lambda=\{\Lambda_0,\Lambda_1,\Lambda_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $\beta>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $\Lambda$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $\Lambda$ suffices to recover (almost surely) either (i) $\beta$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators. Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $\Lambda$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions.