Non-Hermitian ensembles

This article discusses the three-fold family of Ginibre random matrix ensembles (complex, real, and quaternion real) and their elliptic deformations. It also considers eigenvalue correlations that are exactly reduced to two-point kernels in the strongly and weakly non-Hermitian limits of large matrix size. Ginibre introduced the complex, real, and quaternion real random matrix ensembles as a mathematical extension of Hermitian random matrix theory. Statistics of complex eigenvalues are now used in modelling a wide range of physical phenomena. After providing an overview of the complex Ginibre ensemble, the article describes random contractions and the complex elliptic ensemble. It then examines real and quaternion-real Ginibre ensembles, along with real and quaternion-real elliptic ensembles. In particular, it analyses the kernel in the elliptic case as well as the limits of strong and weak non-Hermiticity.