Random normal matrices in the almost-circular regime

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow “band” around the unit circle of width proportional 1∕n, where n is size matrices. For general radially symmetric potentials with various boundary conditions, we derive scaling limits correlation functions, some which appear in previous literature notably context almost-Hermitian also obtain that fluctuations maximal and minimal modulus ensembles follow Gumbel or exponential law depending on conditions.