Abstract: As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density, and in the off-diagonal case, where the Faddeeva plasma kernel emerges. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in higher dimensional determinantal point processes, defined on \mathbb C^d. The models under consideration concern higher dimensional generalizations of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. Their average density of points converges to a uniform law on a 2d-dimensional hyperellipsoid. It is on the boundary of this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of my knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. Based on arXiv:2208.12676