Gaussian fluctuations and moderate deviations of eigenvalues in unitary invariant ensembles

We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.