  In this paper we extend the orthogonal polynomials approach for extreme value
calculations of Hermitian random matrices, developed by Nadal and Majumdar
[1102.0738], to normal random matrices and 2D Coulomb gases in general.
Firstly, we show that this approach provides an alternative derivation of
results in the literature. More precisely, we show convergence of the rescaled
eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel
distribution, as well as universality for an arbitrary radially symmetric
potential. Secondly, it is shown that this approach can be generalised to
obtain convergence of the eigenvalue with smallest modulus and its universality
for ring distributions. Most interestingly, the here presented techniques are
used to compute all slowly varying finite N correction of the above
distributions, which is important for practical applications, given the slow
convergence. Another interesting aspect of this work is the fact that we can
use standard techniques from Hermitian random matrices to obtain the extreme
value statistics of non-Hermitian random matrices resembling the large N
expansion used in context of the double scaling limit of Hermitian matrix
models in string theory.
