The stochastic Airy operator at large temperature
Year: 2019
Type: preprint
Abstract: It was shown in [J. A. Ramírez, B. Rider and B. Virág. J. Amer. Math. Soc. 24, 919-944 (2011)] 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 $1/β$ 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.
Source: arXiv (Cornell University)
Authors: Dumaz, Laure, Labbé, Cyril
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4311560863