Pub Date : 2019-12-29DOI: 10.1142/s2010326321500404
Jennifer Bryson, R. Vershynin, Hongkai Zhao
We prove the Marchenko–Pastur law for the eigenvalues of [Formula: see text] sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the [Formula: see text] coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order [Formula: see text], i.e. the coordinates of the data are all [Formula: see text] different products of [Formula: see text] variables chosen from a set of [Formula: see text] independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is [Formula: see text], and for the random tensor model as long as [Formula: see text]. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.
{"title":"Marchenko–Pastur law with relaxed independence conditions","authors":"Jennifer Bryson, R. Vershynin, Hongkai Zhao","doi":"10.1142/s2010326321500404","DOIUrl":"https://doi.org/10.1142/s2010326321500404","url":null,"abstract":"We prove the Marchenko–Pastur law for the eigenvalues of [Formula: see text] sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the [Formula: see text] coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order [Formula: see text], i.e. the coordinates of the data are all [Formula: see text] different products of [Formula: see text] variables chosen from a set of [Formula: see text] independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is [Formula: see text], and for the random tensor model as long as [Formula: see text]. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90505725","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}
Pub Date : 2019-12-24DOI: 10.1142/s2010326321500258
Elizabeth Meckes, M. Meckes
We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of [Formula: see text] matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for these traces of powers, which imply central limit theorems for polynomial linear eigenvalue statistics. In contrast to the usual situation in random matrix theory, in our approach general, nonnormal matrices turn out to be easier to study than Hermitian matrices.
{"title":"Fluctuations of the spectrum in rotationally invariant random matrix ensembles","authors":"Elizabeth Meckes, M. Meckes","doi":"10.1142/s2010326321500258","DOIUrl":"https://doi.org/10.1142/s2010326321500258","url":null,"abstract":"We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of [Formula: see text] matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for these traces of powers, which imply central limit theorems for polynomial linear eigenvalue statistics. In contrast to the usual situation in random matrix theory, in our approach general, nonnormal matrices turn out to be easier to study than Hermitian matrices.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"35 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75585720","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}
Pub Date : 2019-11-20DOI: 10.1142/s2010326321500210
P. Loubaton, D. Tieplova
The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support [Formula: see text] is characterized. It is also established that the squared singular values are almost surely located in a neighborhood of [Formula: see text].
{"title":"On the behavior of large empirical autocovariance matrices between the past and the future","authors":"P. Loubaton, D. Tieplova","doi":"10.1142/s2010326321500210","DOIUrl":"https://doi.org/10.1142/s2010326321500210","url":null,"abstract":"The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support [Formula: see text] is characterized. It is also established that the squared singular values are almost surely located in a neighborhood of [Formula: see text].","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"32 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87308277","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}
Pub Date : 2019-11-18DOI: 10.1142/s2010326322500526
B. Collins, P. Lamarre, C. Male
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.
{"title":"Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States","authors":"B. Collins, P. Lamarre, C. Male","doi":"10.1142/s2010326322500526","DOIUrl":"https://doi.org/10.1142/s2010326322500526","url":null,"abstract":"In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. \u0000As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"16 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79277606","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}
{"title":"From the totally asymmetric simple exclusion\u0000 process to the KPZ","authors":"J. Quastel, K. Matetski","doi":"10.1090/pcms/026/06","DOIUrl":"https://doi.org/10.1090/pcms/026/06","url":null,"abstract":"","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"7 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73526751","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}
These lectures cover three loosely related topics in random matrix theory. First we discuss the techniques used to bound the least singular value of (nonHermitian) random matrices, focusing particularly on the matrices with jointly independent entries. We then use these bounds to obtain the circular law for the spectrum of matrices with iid entries of finite variance. Finally, we discuss the Lindeberg exchange method which allows one to demonstrate universality of many spectral statistics of matrices (both Hermitian and non-Hermitian). 1. The least singular value This section1 of the lecture notes is concerned with the behaviour of the least singular value σn(M) of an n × n matrix M (or, more generally, the least nontrivial singular value σp(M) of a n×p matrix with p 6 n). This quantity controls the invertibility of M. Indeed, M is invertible precisely when σn(M) is non-zero, and the `2 operator norm ‖M‖op of M−1 is given by 1/σn(M). This quantity is also related to the condition number σ1(M)/σn(M) = ‖M‖op‖M‖op of M, which is of importance in numerical linear algebra. As we shall see in Section 2, the least singular value of M (and more generally, of the shifts 1 √ n M− zI for complex z) will be of importance in rigorously establishing the circular law for iid random matrices M. The least singular value2 σn(M) = inf ‖x‖=1 ‖Mx‖, which sits at the “hard edge” of the spectrum, bears a superficial similarity to the operator norm ‖M‖op = σ1(M) = sup ‖x‖=1 ‖Mx‖ 2010 Mathematics Subject Classification. Primary 60B20; Secondary 60F17.
{"title":"Least singular value, circular law, and\u0000 Lindeberg exchange","authors":"T. Tao","doi":"10.1090/pcms/026/10","DOIUrl":"https://doi.org/10.1090/pcms/026/10","url":null,"abstract":"These lectures cover three loosely related topics in random matrix theory. First we discuss the techniques used to bound the least singular value of (nonHermitian) random matrices, focusing particularly on the matrices with jointly independent entries. We then use these bounds to obtain the circular law for the spectrum of matrices with iid entries of finite variance. Finally, we discuss the Lindeberg exchange method which allows one to demonstrate universality of many spectral statistics of matrices (both Hermitian and non-Hermitian). 1. The least singular value This section1 of the lecture notes is concerned with the behaviour of the least singular value σn(M) of an n × n matrix M (or, more generally, the least nontrivial singular value σp(M) of a n×p matrix with p 6 n). This quantity controls the invertibility of M. Indeed, M is invertible precisely when σn(M) is non-zero, and the `2 operator norm ‖M‖op of M−1 is given by 1/σn(M). This quantity is also related to the condition number σ1(M)/σn(M) = ‖M‖op‖M‖op of M, which is of importance in numerical linear algebra. As we shall see in Section 2, the least singular value of M (and more generally, of the shifts 1 √ n M− zI for complex z) will be of importance in rigorously establishing the circular law for iid random matrices M. The least singular value2 σn(M) = inf ‖x‖=1 ‖Mx‖, which sits at the “hard edge” of the spectrum, bears a superficial similarity to the operator norm ‖M‖op = σ1(M) = sup ‖x‖=1 ‖Mx‖ 2010 Mathematics Subject Classification. Primary 60B20; Secondary 60F17.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"16 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83136228","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}
These are notes to a four-lecture minicourse given at the 2017 PCMI Summer Session on Random Matrices. We give a quick introduction to the theory of large random matrices by taking limits that preserve their operator structure, rather than just their eigenvalues. The operator structure takes the role of exact formulas, and allows for results in the context of general β-ensembles. Along the way, we cover a non-computational proof of the Wiegner semicircle law, a quick proofs for the Füredi-Komlós result on the top eigenvalue, as well as the BBP phase transition. 1. The Gaussian Ensembles 1.1. The Gaussian Orthogonal and Unitary Ensembles. One of the earliest appearances of random matrices in mathematics was due to Eugene Wigner in the 1950’s. Let G be an n×nmatrix with independent standard normal entries. Then Mn = G+Gt √ 2 . This distribution on symmetric matrices is called the Gaussian Orthogonal Ensemble, because it is invariant under orthogonal conjugation. For any orthogonal matrix OMnO has the same distribution as Mn. To check this, note that OG has the same distribution as G be the rotation invariance of the Gaussian column vectors, and the same is true for OGO−1 by the rotation invariance of the row vectors. To finish note that orthogonal conjugation commutes with symmetrization. If we instead start with a matrix with independent standard complex Gaussian entries, we get the Gaussian Unitary ensemble. To see how the eigenvalues behave, we recall the following classical theorem. Theorem 1.1.1. Suppose Mn has GOE or GUE distribution then Mn has eigenvalue density (1.1.2) f(λ1, ..., λn) = 1 Zn n ∏
{"title":"A short introduction to operator limits of\u0000 random matrices","authors":"D. Holcomb, B. Virág","doi":"10.1090/pcms/026/05","DOIUrl":"https://doi.org/10.1090/pcms/026/05","url":null,"abstract":"These are notes to a four-lecture minicourse given at the 2017 PCMI Summer Session on Random Matrices. We give a quick introduction to the theory of large random matrices by taking limits that preserve their operator structure, rather than just their eigenvalues. The operator structure takes the role of exact formulas, and allows for results in the context of general β-ensembles. Along the way, we cover a non-computational proof of the Wiegner semicircle law, a quick proofs for the Füredi-Komlós result on the top eigenvalue, as well as the BBP phase transition. 1. The Gaussian Ensembles 1.1. The Gaussian Orthogonal and Unitary Ensembles. One of the earliest appearances of random matrices in mathematics was due to Eugene Wigner in the 1950’s. Let G be an n×nmatrix with independent standard normal entries. Then Mn = G+Gt √ 2 . This distribution on symmetric matrices is called the Gaussian Orthogonal Ensemble, because it is invariant under orthogonal conjugation. For any orthogonal matrix OMnO has the same distribution as Mn. To check this, note that OG has the same distribution as G be the rotation invariance of the Gaussian column vectors, and the same is true for OGO−1 by the rotation invariance of the row vectors. To finish note that orthogonal conjugation commutes with symmetrization. If we instead start with a matrix with independent standard complex Gaussian entries, we get the Gaussian Unitary ensemble. To see how the eigenvalues behave, we recall the following classical theorem. Theorem 1.1.1. Suppose Mn has GOE or GUE distribution then Mn has eigenvalue density (1.1.2) f(λ1, ..., λn) = 1 Zn n ∏","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"32 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88413283","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}
{"title":"The semicircle law and beyond: The shape of\u0000 spectra of Wigner matrices","authors":"Ioana Dumitriu","doi":"10.1090/pcms/026/02","DOIUrl":"https://doi.org/10.1090/pcms/026/02","url":null,"abstract":"","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"114 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75864297","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}
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