Almost-Hermitian random matrices: eigenvalue density in the complex plane
Abstract
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 N TrÂs2 is of the same order as that of TrĤ2 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 distribuyion 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”.
- Publication:
-
Physics Letters A
- Pub Date:
- February 1997
- DOI:
- 10.1016/S0375-9601(96)00904-8
- arXiv:
- arXiv:cond-mat/9606173
- Bibcode:
- 1997PhLA..226...46F
- Keywords:
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- Condensed Matter;
- High Energy Physics - Lattice;
- High Energy Physics - Theory;
- Nonlinear Sciences - Chaotic Dynamics
- E-Print:
- 9 pages, 1 figure in eps format, LaTeX, submitted to Journ Phys A