Universality classes of non-Hermitian random matrices

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.