CEREMADE, Université Paris-Dauphine
Date(s) : 18/10/2019 iCal
11 h 00 min - 12 h 00 min
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 ∞: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on ℝ of intensity exdx 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.