Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane

We consider an ensemble of large non-Hermitian random matrices of the form Ĥ + iÂs, where Ĥ and Âs are Hermitian statistically independent random N × N matrices. We demonstrate the existence of a new nontrivial regime of weak non-Hermiticity characterized by the condition that the average of NTrÂs is of the same order as that of TrĤ 2 when N → ∞. We find explicitly the density of complex eigenvalues for this regime in the limit of infinite matrix dimension. The density determines the eigenvalue distribution in the crossover regime between random Hermitian matrices whose real eigenvalues are distributed according to the Wigner semi-circle law and random complex matrices whose eigenvalues are distributed in the complex plane according to the so-called “elliptic law”. Recently there was a growth of interest in statistics of complex eigenvalues of large random matrices, both in physical and mathematical literature (see Refs. 1 20). Non-Hermitian random Hamiltonians appear naturally when one deals with quantum scattering problems in open chaotic systems [7,8,11,15,16,18], studies a motion of flux lines in superconductors with columnar defects [19] or is interested in chiral symmetry breaking in quantum chromodynamics [20]. Complex random matrices appear also in studies of dissipative quantum maps [6,9]. On the other hand, closely related ensembles of real asymmetric random matrices enjoy applications in neural network dynamics [5,12] and in the problem of the localization transition of random heteropolymer chains [21]. The actual progress in understanding the properties of random matrices with complex eigenvalues is rather limited, however. Most of the known results refer to the case of strong non-Hermiticity or asymmetry. Namely, they deal with those types of matrices for which ∗On leave from Petersburg Nuclear Physics Institute, Gatchina 188350, Russia †On leave from B.I. Verkin Institute for Low Temperature Physics, Kharkov, 310164, Ukraine.