The stochastic Airy operator at large temperature. (English) Zbl 1504.35204

Dumaz, Laure; Labbé, Cyril

Ann. Appl. Probab. 32, No. 6, 4481-4534 (2022).

Summary: It was shown in [J. A. Ramírez et al., J. Am. Math. Soc. 24, No. 4, 919–944 (2011; Zbl 1239.60005)] 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 R of intensity \({e^x}\mathrm{d}x\) 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.

MSC:

35P20	Asymptotic distributions of eigenvalues in context of PDEs
35R60	PDEs with randomness, stochastic partial differential equations
60H25	Random operators and equations (aspects of stochastic analysis)
60J60	Diffusion processes

Keywords:

beta ensemble; diffusion; localization; Riccati transform; stochastic Airy operator

Citations:

Zbl 1239.60005

References:

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