On the Semicircular Law of Large-Dimensional Random Quaternion Matrices

Yanqing Yin (), Zhidong Bai () and Jiang Hu ()
Additional contact information
Yanqing Yin: Northeast Normal University
Zhidong Bai: Northeast Normal University
Jiang Hu: Northeast Normal University

Journal of Theoretical Probability, 2016, vol. 29, issue 3, 1100-1120

Abstract: Abstract It is well known that the Gaussian symplectic ensemble is defined on the space of $$n\times n$$ n × n quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.

Keywords: GSE; Quaternion matrices; Semicircular law; Primary 15B52; 60F15; 62E20; Secondary 60F17 (search for similar items in EconPapers)
Date: 2016
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-015-0610-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:29:y:2016:i:3:d:10.1007_s10959-015-0610-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-015-0610-3

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().