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Zeng, Xingyuan; Hou, Zhenting

Precise asymptotics for beta ensembles. (English) Zbl 1276.15020

Indian J. Pure Appl. Math. 44, No. 3, 383-404 (2013).
The edge properties of the spectrum of random matrices are investigated. The paper extends results by Z. Su [Acta Math. Sin., Engl. Ser. 24, No. 6, 971–982 (2008; Zbl 1154.60034)]) who first obtained precise asymptotic results for the largest eigenvalues of the Gaussian unitary ensemble and of the Laguerre unitary ensemble for \(\beta = 2\). The new results are obtained for both the largest eigenvalues and the smallest eigenvalues of the \(\beta\)-Hermite and \(\beta\)-Laguerre ensembles using the general \(\beta\) Tracy-Widom law together with the results for small deviation inequalities for random matrices in the paper by M. Ledoux and B. Rider [Electron. J. Probab. 15, 1319–1343 (2010; Zbl 1228.60015)].
Reviewer: Václav Burjan (Praha)

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A42 Inequalities involving eigenvalues and eigenvectors

Keywords:

extremal eigenvalues; \(\beta\)-ensembles; general \(\beta\) Tracy-Widom law; small deviation inequalities; random matrices; Gaussian unitary ensemble; Laguerre unitary ensemble

Citations:

Zbl 1154.60034; Zbl 1228.60015
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References:

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