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We address the question of how the celebrated universality of localcorrelations for the real eigenvalues of Hermitian random matrices of size NxNcan be extended to complex eigenvalues in the case of random matrices withoutsymmetry. Depending on the location in the spectrum, particular large-N limits(the so-called weakly non-Hermitian limits) lead to one-parameter deformationsof the Airy, sine and Bessel kernels into the complex plane. This makes theiruniversality highly suggestive for all symmetry classes. We compare all theknown limiting real kernels and their deformations into the complex plane forall three Dyson indices beta=1,2,4, corresponding to real, complex andquaternion real matrix elements. This includes new results for Airy kernels inthe complex plane for beta=1,4. For the Gaussian ensembles of elliptic Ginibreand non-Hermitian Wishart matrices we give all kernels for finite N, built fromorthogonal and skew-orthogonal polynomials in the complex plane. Finally wecomment on how much is known to date regarding the universality of thesekernels in the complex plane, and discuss some open problems.

Universality Conjecture for all Airy, Sine and Bessel Kernels in the Complex Plane