A Classification Ofnon-hermitian Random Matrices

We present a classification of non-hermitian random matrices based on implementing commuting discrete symmetries. It contains 43 classes. This generalizes the classification of hermitian random matrices due to Altland-Zirnbauer and it also extends the Ginibre ensembles of nonhermitian matrices [1]. Random matrix theory originates from the work of Wigner and Dyson on random hamiltonians [2]. Since then it has been applied to a large variety of problems ranging from enumerative topology, combinatorics, to localization phenomena, fluctuating surfaces, integrable or chaotic systems, etc... Non-hermitian random matrices also have applications to interesting quantum problems such as open choatic scattering, dissipative quantum maps, non-hermitian localization, etc... See e.g. ref.[6] for an introduction. The aim of this short note is to extend the Dyson [2] and Altland-Zirnbauer [4] classifications of hermitian random matrix ensembles to the non-hermitian ones. 1. What are the rules? As usual, randommatrix ensembles are constructed by selecting classes of matrices with specified properties under discrete symmetries [2, 3]. To define these ensembles we have to specify (i) what are the discrete symmetries, (ii) what are the equivalence relations among the matrices, and (iii) what are the probablility measures for each class.