THE STOCHASTIC AIRY OPERATOR AT LARGE TEMPERATURE

• 成果类型: Article
• 署名作者: Dumaz, Laure; Labbe, Cyril
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1793
• 发表日期: 2022
• 页码: 4481-4534
• 关键词: beta ensembles

摘要

It was shown in (J. Amer. Math. Soc. 24 (2011) 919-944) 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 infinity: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on R of intensity ex dx 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.