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The two-dimensional one-component plasma is an ubiquitous model for severalvortex systems. For special values of the coupling constant $\beta q^2$ (where$q$ is the particles charge and $\beta$ the inverse temperature), the modelalso 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 thestatistics of a class of radial observables produces a rich phase diagram, andtheir asymptotic behaviour in terms of large deviation functions is calculatedexplicitly, including next-to-leading terms up to order 1/N. We demonstrate asplit-off phenomenon associated to atypical fluctuations of the edge densityprofile. We also show explicitly that a failure of the fluid phase assumptionof the plasma can break a genuine $1/N$-expansion of the free energy. Ourfindings are corroborated by numerical comparisons with exact finite-N formulaevalid for $\beta q^2=2$.

Large deviations of radial statistics in the two-dimensional one-component plasma