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

Author

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  • Cambyse Pakzad

    (Université Paris Descartes)

Abstract

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

Suggested Citation

  • Cambyse Pakzad, 2020. "Large Deviations Principle for the Largest Eigenvalue of the Gaussian $$\beta $$β-Ensemble at High Temperature," Journal of Theoretical Probability, Springer, vol. 33(1), pages 428-443, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-019-00882-4
    DOI: 10.1007/s10959-019-00882-4
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