  The two-dimensional one-component plasma is an ubiquitous model for several
vortex systems. For special values of the coupling constant $\beta q^2$ (where
$q$ is the particles charge and $\beta$ the inverse temperature), the model
also corresponds to the eigenvalues distribution of normal matrix models.
Several features of the system are discussed in the limit of large number $N$
of particles for generic values of the coupling constant. We show that the
statistics of a class of radial observables produces a rich phase diagram, and
their asymptotic behaviour in terms of large deviation functions is calculated
explicitly, including next-to-leading terms up to order 1/N. We demonstrate a
split-off phenomenon associated to atypical fluctuations of the edge density
profile. We also show explicitly that a failure of the fluid phase assumption
of the plasma can break a genuine $1/N$-expansion of the free energy. Our
findings are corroborated by numerical comparisons with exact finite-N formulae
valid for $\beta q^2=2$.
