  A famous result going back to Eric Kostlan states that the moduli of the
eigenvalues of random normal matrices with radial potential are independent yet
non identically distributed. This phenomenon is at the heart of the asymptotic
analysis of the edge, and leads in particular to the Gumbel fluctuation of the
spectral radius when the potential is quadratic. In the present work, we show
that a wide variety of laws of fluctuation are possible, beyond the already
known cases, including for instance Gumbel and exponential laws at unusual
speeds. We study the convergence in law of the spectral radius as well as the
limiting point process at the edge. Our work can also be seen as the asymptotic
analysis of the edge of two-dimensional determinantal Coulomb gases and the
identification of the limiting kernels.
