Fine Asymptotic Behavior for Eigenvalues of Random Normal Matrices: Ellipse Case

We consider the random normal matrices with quadratic external potentials where the associated orthogonal polynomials are Hermite polynomials and the limiting support (called droplet) of the eigenvalues is an ellipse. We calculate the density of the eigenvalues near the boundary of the droplet up to the second subleading corrections and express the subleading corrections in terms of the curvature of the droplet boundary. From this result, we additionally get the expected number of eigenvalues outside the droplet. We also show that a certain Cauchy transform of the orthogonal polynomial vanishes in the bulk of the droplet up to an exponentially small error