  We consider determinantal Coulomb gas ensembles with a class of discrete
rotational symmetric potentials whose droplets consist of several disconnected
components. Under the insertion of a point charge at the origin, we derive the
asymptotic behaviour of the correlation kernels both in the macro- and
microscopic scales. In the macroscopic scale, this particularly shows that
there are strong correlations among the particles on the boundary of the
droplets. In the microscopic scale, this establishes the edge universality. For
the proofs, we use the nonlinear steepest descent method on the matrix
Riemann-Hilbert problem to derive the asymptotic behaviours of the associated
planar orthogonal polynomials and their norms up to the first subleading terms.
