Large scale correlations in normal non-Hermitian matrix ensembles

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with a general non-Gaussian measure and in ensembles of general non-Hermitian matrices with a class of non-Gaussian measures. In both cases, the eigenvalues are complex and in the large N limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N, in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues.