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

Cambyse Pakzad ()
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Cambyse Pakzad: Université Paris Descartes

Journal of Theoretical Probability, 2020, vol. 33, issue 1, 428-443

Abstract: 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.

Keywords: Large deviations principle; Random matrices; Gaussian $$\beta $$ β -ensembles; Extreme eigenvalue; 60B20; 60F10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00882-4

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