Random Matrices-Theory and Applications最新文献_第6页

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations 不同相关性的样本协方差矩阵相关模型的经验谱分布

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2021-03-04 DOI: 10.1142/s2010326322500307

Alicja Dembczak-Kołodziejczyk, A. Lytova

{"title":"On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations","authors":"Alicja Dembczak-Kołodziejczyk, A. Lytova","doi":"10.1142/s2010326322500307","DOIUrl":"https://doi.org/10.1142/s2010326322500307","url":null,"abstract":"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].","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83652553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 5

The limit empirical spectral distribution of complex matrix polynomials 复矩阵多项式的极限经验谱分布

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2021-02-03 DOI: 10.1142/S201032632250023X

Giovanni Barbarino, V. Noferini

{"title":"The limit empirical spectral distribution of complex matrix polynomials","authors":"Giovanni Barbarino, V. Noferini","doi":"10.1142/S201032632250023X","DOIUrl":"https://doi.org/10.1142/S201032632250023X","url":null,"abstract":"We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77629061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Additivity violation of quantum channels via strong convergence to semi-circular and circular elements 半圆和圆元强收敛量子通道的可加性破坏

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2021-01-02 DOI: 10.1142/s2010326322500125

M. Fukuda, Takahiro Hasebe, Shinya Sato

引用次数: 1

Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices 由大维随机矩阵特征向量定义的随机函数集合的弱收敛性

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-12-23 DOI: 10.1142/s2010326322500332

J. W. Silverstein

{"title":"Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices","authors":"J. W. Silverstein","doi":"10.1142/s2010326322500332","DOIUrl":"https://doi.org/10.1142/s2010326322500332","url":null,"abstract":"For each n, let Un be Haar distributed on the group of n × n unitary matrices. Let xn,1, . . . ,xn,m denote orthogonal nonrandom unit vectors in C n and let un,k = (uk, . . . , u n k) ∗ = U∗ nxn,k, k = 1, . . . , m. Define the following functions on [0,1]: X k,k n (t) = √ n ∑[nt] i=1(|uk|− 1 n ),X ′ n (t) = √ 2n ∑[nt] i=1 ū i ku i k′ , k < k ′. Then it is proven thatX n ,RXk,k ′ n , IXk,k′ n , considered as random processes in D[0, 1], converge weakly, as n → ∞, to m independent copies of Brownian bridge. The same result holds for the m(m + 1)/2 processes in the real case, where On is real orthogonal Haar distributed and xn,i ∈ R, with √ n in X n and √ 2n in X ′ n replaced with √ n 2 and √ n, respectively. This latter result will be shown to hold for the matrix of eigenvectors of Mn = (1/s)VnV T n where Vn is n × s consisting of the entries of {vij}, i, j = 1, 2, . . . , i.i.d. standardized and symmetrically distributed, with each xn,i = {±1/ √ n, . . . ,±1/√n}, and n/s→ y > 0 as n→ ∞. This result extends the result in J.W. Silverstein Ann. Probab. 18 1174-1194. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix Bn = θvnv ∗ n + Sn is studied where Sn is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or Sn = Mn, θ > 0 nonrandom, and vn is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to vn with the eigenvector associated with the largest eigenvalue of Bn.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79705275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Pair dependent linear statistics for CβE CβE的对相关线性统计量

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-12-17 DOI: 10.1142/S2010326321500350

A. Aguirre, A. Soshnikov, Joshua Sumpter

引用次数: 6

CLT with explicit variance for products of random singular matrices related to Hill’s equation 与希尔方程相关的随机奇异矩阵乘积的显方差CLT

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-12-03 DOI: 10.1142/S2010326322500186

Phanuel Mariano, Hugo Panzo

引用次数: 0

Gap probabilities in the bulk of the Airy process 艾里过程的间隙概率

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-12-02 DOI: 10.1142/s2010326322500228

E. Blackstone, C. Charlier, J. Lenells

引用次数: 5

Some patterned matrices with independent entries 一些具有独立元素的模式矩阵

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-11-27 DOI: 10.1142/S2010326321500301

A. Bose, Koushik Saha, Priyanka Sen

引用次数: 5

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble β = 2 stieltje - wigert随机矩阵系综的全局和局部尺度极限

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-11-23 DOI: 10.1142/s2010326322500204

P. Forrester

{"title":"Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble","authors":"P. Forrester","doi":"10.1142/s2010326322500204","DOIUrl":"https://doi.org/10.1142/s2010326322500204","url":null,"abstract":"The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72428028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 13

On Random Matrices Arising in Deep Neural Networks: General I.I.D. Case 深度神经网络中产生的随机矩阵:一般I.I.D.案例

IF 0.9 4区数学

Random Matrices-Theory and Applications Pub Date : 2020-11-20 DOI: 10.1142/s2010326322500460

L. Pastur, V. Slavin

{"title":"On Random Matrices Arising in Deep Neural Networks: General I.I.D. Case","authors":"L. Pastur, V. Slavin","doi":"10.1142/s2010326322500460","DOIUrl":"https://doi.org/10.1142/s2010326322500460","url":null,"abstract":"We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability has to be justified. The justification has been given in [22] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper we use another, more streamlined, version of the techniques of random matrix theory to generalize the results of [22] to the case where the entries of the synaptic weight matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73028962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 9