The stochastic Airy operator at large temperature

Abstract

It was shown in (J. Amer. Math. Soc. 24 (2011) 919–944) that the edge of the spectrum of β ensembles converges in the large N limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/\mathit{\beta }$ goes to ∞: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on R of intensity ${\mathit{e}}^{\mathit{x}}\mathit{d}\mathit{x}$ and that the eigenfunctions converge to Dirac masses centered at i.i.d. points with exponential laws. Furthermore, we obtain a precise description of the microscopic behavior of the eigenfunctions near their localization centers.