Precise asymptotics for beta ensembles (Q358970)

The edge properties of the spectrum of random matrices are investigated. The paper extends results by \textit{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 \textit{M. Ledoux} and \textit{B. Rider} [Electron. J. Probab. 15, 1319--1343 (2010; Zbl 1228.60015)].