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Article Dans Une Revue The Annals of Applied Probability Année : 2022

The stochastic Airy operator at large temperature

Résumé

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.

Dates et versions

hal-02292731 , version 1 (20-09-2019)

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Citer

Laure Dumaz, Cyril Labbé. The stochastic Airy operator at large temperature. The Annals of Applied Probability, 2022, 32 (6), ⟨10.1214/22-AAP1793⟩. ⟨hal-02292731⟩
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