For beta-ensembles with convex polynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.
"Near-extreme eigenvalues in the beta-ensembles." Electron. J. Probab. 21 1 - 17, 2016. https://doi.org/10.1214/16-EJP4