The notion of an orthogonal polynomial ensemble generalizes many
important point processes arising in random matrix theory, probability
and combinatorics.
This talk describes recent joint work with Maurice Duits dealing with
the fluctuations of the random empirical measure for general
orthogonal polynomial ensembles, on all scales, for both varying and
fixed measures.
We obtain a general concentration inequality and prove both global
(`macroscopic') and local (`mesoscopic') almost sure convergence of
linear statistics under fairly weak assumptions on the ensemble. An
important role in the analysis is played by a strengthening of the
Nevai condition from the theory of orthogonal polynomials.
No previous knowledge of orthogonal polynomial ensembles or orthogonal
polynomial theory is assumed.