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.

scholarly journals Explicit generating series for connection coefficients

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Functions ◽  
Double Coset ◽  
Type A ◽  
Explicit Computation ◽  
Zonal Polynomials ◽  
Connection Coefficients ◽  
Generating Series ◽  
International Audience ◽  
Commutative Subalgebras ◽  
Orientable Surfaces

International audience This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type $[a,b,1^{n-a-b}]$. Cet article est dédié au calcul explicite des séries génératrices des constantes de structure de deux sous-algèbres commutatives de l'algèbre de groupe du groupe symétrique, l'algèbre de classes et l'algèbre de double classe latérale. Tel que montrè par Hanlon, Stanley and Stembridge (1992), ces séries déterminent la distribution spectrale de certaines matrices aléatoires importantes en statistique. Morales et Vassilieva (2009, 2011) ont trouvè des formules explicites pour ces séries génératrices en termes des monômes symétriques en introduisant une bijection entre les hypercartes partitionnées sur des surfaces (localement) orientables et certains arbres et forêts décorées. Grâce à des moyens purement algébriques, nous retrouvons la formule pour l'algèbre de classe et déterminons une nouvelle formule plus simple pour l'algèbre de double classe latérale. En tant que point saillant de notre démonstration nous calculons une formulation explicite pour les polynômes zonaux indexés par des partitions de type $[a,b,1^{n-a-b}]$.

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 On an Identity due to Bump and Diaconis, and Tracy and Widom

2011 ◽  
Vol 54 (2) ◽  
pp. 255-269 ◽  
Author(s):  
Paul-Olivier Dehaye
Keyword(s):  
Differential Operator ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Finite Rank ◽  
Symmetric Functions ◽  
Matrix Theory ◽  
Laguerre Polynomials ◽  
Symmetric Groups ◽  
Vertex Operators ◽  
Rank One

AbstractA classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

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.

Sign in / Sign up

Export Citation Format

Share Document