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On the Semicircular Law of Large-Dimensional Random Quaternion Matrices
- Source :
- Journal of Theoretical Probability. 29:1100-1120
- Publication Year :
- 2015
- Publisher :
- Springer Science and Business Media LLC, 2015.
-
Abstract
- It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) 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.
- Subjects :
- Statistics and Probability
Lemma (mathematics)
Pure mathematics
General Mathematics
Gaussian
010102 general mathematics
01 natural sciences
Hermitian matrix
010104 statistics & probability
symbols.namesake
Law
symbols
Multiplication
0101 mathematics
Statistics, Probability and Uncertainty
Quaternion
Commutative property
Eigenvalues and eigenvectors
Symplectic geometry
Mathematics
Subjects
Details
- ISSN :
- 15729230 and 08949840
- Volume :
- 29
- Database :
- OpenAIRE
- Journal :
- Journal of Theoretical Probability
- Accession number :
- edsair.doi...........ff0984ae47a0bfbfe028bfc65228c199