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

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.

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Correspondence to Cambyse Pakzad.

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Pakzad, C. Large Deviations Principle for the Largest Eigenvalue of the Gaussian \(\beta \)-Ensemble at High Temperature. J Theor Probab 33, 428–443 (2020). https://doi.org/10.1007/s10959-019-00882-4

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Keywords

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

Mathematics Subject Classification (2010)

  • 60B20
  • 60F10