We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately.
"Small Deviations for Beta Ensembles." Electron. J. Probab. 15 1319 - 1343, 2010. https://doi.org/10.1214/EJP.v15-798