The stochastic Airy operator at large temperature

TL;DR

This paper studies the spectral behavior of the stochastic Airy operator at high temperature, revealing a transition to Poisson statistics and localized eigenfunctions, extending understanding of spectral limits in random matrix theory.

Contribution

It provides a complete description of the eigenvalue and eigenfunction behavior of the stochastic Airy operator as temperature increases to infinity, including convergence to Poisson processes and localization details.

Findings

Eigenvalues form a Poisson process with intensity e^x dx at high temperature.

Eigenfunctions localize at IID points with exponential distribution.

Microscopic eigenfunction behavior near localization centers is characterized.

Abstract

It was shown in [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $\beta$ 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 $1/\beta$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R}$ of intensity $e^x dx$ and that the eigenfunctions converge to Dirac masses centered at IID points with exponential laws. Furthermore, we obtain a precise description of the microscopic behavior of the eigenfunctions near their localization centers.