Burda, Zdzislaw ORCID: 0000-0002-9656-9570 and Swiech, Artur (2015). Quaternionic R transform and non-Hermitian random matrices. Phys. Rev. E, 92 (5). COLLEGE PK: AMER PHYSICAL SOC. ISSN 1550-2376
Full text not available from this repository.Abstract
Using the Cayley-Dickson construction we rephrase and review the non-Hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp, and I. Zahed, Nucl. Phys. B 501, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-Hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (noncrossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its Hermitian conjugate X-dagger: << 1/N (TrXXXc)-X-a-X-dagger b ...>> for N -> infinity. We show that the R transform for Gaussian elliptic laws is given by a simple linear quaternionic map R(z + wj) = x + sigma(2)(mu e(2i phi)z + wj) where (z, w) is the Cayley-Dickson pair of complex numbers forming a quaternion q = (z, w) = z + wj. This map has five real parameters Rex, Imx, phi, sigma, and mu. We use the R transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.
Item Type: | Journal Article | ||||||||||||
Creators: |
|
||||||||||||
URN: | urn:nbn:de:hbz:38-387120 | ||||||||||||
DOI: | 10.1103/PhysRevE.92.052111 | ||||||||||||
Journal or Publication Title: | Phys. Rev. E | ||||||||||||
Volume: | 92 | ||||||||||||
Number: | 5 | ||||||||||||
Date: | 2015 | ||||||||||||
Publisher: | AMER PHYSICAL SOC | ||||||||||||
Place of Publication: | COLLEGE PK | ||||||||||||
ISSN: | 1550-2376 | ||||||||||||
Language: | English | ||||||||||||
Faculty: | Unspecified | ||||||||||||
Divisions: | Unspecified | ||||||||||||
Subjects: | no entry | ||||||||||||
Uncontrolled Keywords: |
|
||||||||||||
URI: | http://kups.ub.uni-koeln.de/id/eprint/38712 |
Downloads
Downloads per month over past year
Altmetric
Export
Actions (login required)
View Item |