  Non-Hermitian random matrices have been utilized in such diverse fields as
dissipative and stochastic processes, mesoscopic physics, nuclear physics, and
neural networks. However, the only known universal level-spacing statistics is
that of the Ginibre ensemble characterized by complex-conjugation symmetry.
Here we report our discovery of two other distinct universality classes
characterized by transposition symmetry. We find that transposition symmetry
alters repulsive interactions between two neighboring eigenvalues and deforms
their spacing distribution. Such alteration is not possible with other
symmetries including Ginibre's complex-conjugation symmetry which can affect
only nonlocal correlations. Our results complete the non-Hermitian counterpart
of Wigner-Dyson's threefold universal statistics of Hermitian random matrices
and serve as a basis for characterizing nonintegrability and chaos in open
quantum systems with symmetry.
