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Large Deviations Principle for the Largest Eigenvalue of the Gaussian \(\beta \)-Ensemble at High Temperature

  • Cambyse PakzadEmail author
Article
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Abstract

We consider the Gaussian \(\beta \)-ensemble when \(\beta \) scales with n (the number of particles) such that \(\displaystyle {{n}^{-1}\ll \beta \ll 1}\). Under a certain regime for \(\beta \), we show that the largest particle satisfies a large deviations principle in \(\mathbb {R}\) with speed \(n\beta \) and explicit rate function. As a consequence, the largest particle converges in probability to 2, the rightmost point of the semicircle law.

Keywords

Large deviations principle Random matrices Gaussian \(\beta \)-ensembles Extreme eigenvalue 

Mathematics Subject Classification (2010)

60B20 60F10 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.MAP 5, UMR CNRS 8145Université Paris DescartesParisFrance

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