Universität zu Köln

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

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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:	Creators Email ORCID ORCID Put Code Burda, Zdzislaw UNSPECIFIED orcid.org/0000-0002-9656-9570 UNSPECIFIED Swiech, Artur UNSPECIFIED UNSPECIFIED UNSPECIFIED
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:	Keywords Language CHAOTIC SCATTERING CROSSOVER; SINGULAR-VALUE; PHASE-SHIFTS; TIME DELAYS; STATISTICS; MODELS; MULTIPLICATION; ENSEMBLES; SPECTRUM; POLES Multiple languages Physics, Fluids & Plasmas; Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/38712