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We investigate a one-parameter family of Coulomb gases in two dimensions,which are confined to an ellipse, due to a hard wall constraint, and aresubject to an additional external potential. At inverse temperature $\beta=2$we can use the technique of planar orthogonal polynomials, borrowed from randommatrix theory, to explicitly determine all $k$-point correlation functions fora fixed number of particles $N$. These are given by the determinant of thekernel of the corresponding orthogonal polynomials, which in our case are theGegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials,depending on the choice of external potential, as shown in a companion paperrecently published by three of the authors. In the rotationally invariant case,when the ellipse becomes the unit disc, our findings agree with that of theensemble of truncated unitary random matrices. The thermodynamical large-$N$limit is investigated in the local scaling regime in the bulk and at the edgeof the spectrum at weak and strong non-Hermiticity. We find new universalityclasses in these limits and recover the sine- and Bessel-kernel in theHermitian limit. The limiting global correlation functions of particles in theinterior of the ellipse are more difficult to obtain but found in the specialcases corresponding to the Chebyshev polynomials.

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality