Conformal Deformation from Normal to Hermitian Random Matrix Ensembles

We investigate the eigenvalues statistics of ensembles of normal random matrices when their order N tends to infinite. In the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We study the conformal deformations of normal random ensembles to Hermitian random ensembles and give sufficient conditions for the latter to be a Wigner ensemble.