On a comparative study of French's spectral distribution and the random matrix theory

1971 ◽  
Vol 164 (3) ◽  
pp. 658-663 ◽  
Author(s):  
Nazakat Ullah
Keyword(s):  
Comparative Study ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory

On the limit of the spectral distribution of large-dimensional random quaternion covariance matrices

2017 ◽  
Vol 06 (02) ◽  
pp. 1750004 ◽  
Author(s):  
Yanqing Yin ◽  
Jiang Hu
Keyword(s):  
Machine Learning ◽  
Image Processing ◽  
Adaptive Filtering ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Moment Condition ◽  
Quaternion Matrices

The use of quaternions and quaternion matrices in practice, such as in machine learning, adaptive filtering, vector sensing and image processing, has recently been rapidly gaining in popularity. In this paper, by applying random matrix theory, we investigate the spectral distribution of large-dimensional quaternion covariance matrices when the quaternion samples are drawn from a population that satisfies a mild moment condition. We also apply the result to several common models.

scholarly journals On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

2021 ◽  
Author(s):  
Alicja Dembczak-Kołodziejczyk ◽  
Anna Lytova
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Random Variable ◽  
Covariance Matrices ◽  
Convergence In Probability ◽  
Stat Phys ◽  
Sample Covariance ◽  
Arbitrary Space

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].

scholarly journals Bijective evaluation of the connection coefficients of the double coset algebra

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alejandro H. Morales ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Symmetric Functions ◽  
Matrix Theory ◽  
Double Coset ◽  
Hyperoctahedral Group ◽  
Connection Coefficients ◽  
Generating Series ◽  
International Audience

International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests. Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory

History—an overview

2018 ◽  
Author(s):  
Oriol Bohigas ◽  
Hans A. Weidenmüller
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Rapid Development ◽  
The Past ◽  
History Of ◽  
And Mathematics

An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.

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