  For beta ensembles with convex poynomial 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.
