AlmostHermitian random matrices: eigenvalue density in the complex plane
Abstract
We consider an ensemble of large nonHermitian random matrices of the form Ĥ + iÂ_{s}, where Ĥ and Â_{s} are Hermitian statistically independent random N × N matrices. We demonstrate the existence of a new nontrivial regime of weak nonHermiticity characterized by the condition that the average of N TrÂ_{s}^{2} 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 distribuyion in the crossover regime between random Hermitian matrices whose real eigenvalues are distributed according to the Wigner semicircle law and random complex matrices whose eigenvalues are distributed in the complex plane according to the socalled “elliptic law”.
 Publication:

Physics Letters A
 Pub Date:
 February 1997
 DOI:
 10.1016/S03759601(96)009048
 arXiv:
 arXiv:condmat/9606173
 Bibcode:
 1997PhLA..226...46F
 Keywords:

 Condensed Matter;
 High Energy Physics  Lattice;
 High Energy Physics  Theory;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 9 pages, 1 figure in eps format, LaTeX, submitted to Journ Phys A