Singularities of the density of states of random Gram matrices

Alt, Johannes

doi:10.1214/17-ecp97

Cited by 18 publications

(33 citation statements)

References 10 publications

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“…Remark Up to notational differences, , are the centered case(

double-struckE M_{n} = 0

) of the equations in (see section 5.1 in ), where a noncentered form of the equations were also derived under the assumptions of (4 + ε )‐bounded moments and the continuity of the variance profile. Recently, , were also studied in , where the local law for the centered case was proved under stronger assumptions including bounded k ‐moments of each entry for each k and irreducibility condition on the variance profile. Our Theorems and give the weakest assumption so far for the existence of the limiting distribution and the quadratic vector equations only for the centered case.…”

Section: Random Gram Matricesmentioning

confidence: 99%

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.

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“…Remark Up to notational differences, , are the centered case(

double-struckE M_{n} = 0

Section: Random Gram Matricesmentioning

confidence: 99%

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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“…In the following we state some fundamental properties of the Gram matrix H and of its resolvent G (for a detailed description see [2] and [3]). Let m 1 , m 2 : H → H be the unique solutions of the system .…”

Section: 2mentioning

confidence: 99%

Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

Cipolloni

Erdős

2019

Random Matrices: Theory Appl.

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We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix [Formula: see text] and its minor [Formula: see text]. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of [Formula: see text] and [Formula: see text]. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.

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“…The self-adjoint linearization method has been proved to be useful in studying the local laws of random matrices of Gram type [1,2,16,18,38,48,49]. We now introduce a generalization of this method, which was introduced in [50] to study the null SCC matrix C XY .…”

Section: Linearization Methods and Local Lawsmentioning

confidence: 99%

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Ma¹,

Yang²

2021

Preprint

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Consider two random vectors r x P R p and r y P R q of the forms rx " Az `C1{2 1 x and r y " Bz `C1{2 2 y, where x P R p , y P R q and z P R r are independent random vectors with i.i.d. entries of zero mean and unit variance, C1 and C2 are p ˆp and q ˆq deterministic population covariance matrices, and A and B are p ˆr and q ˆr deterministic factor loading matrices. With n independent observations of pr x, r yq, we study the sample canonical correlations between rx and r y. We consider the high-dimensional setting with finite rank correlations, that is, p{n Ñ c1 and q{n Ñ c2 as n Ñ 8 for some constants c1 P p0, 1q and c2 P p0, 1 ´c1q, and r is a fixed integer. Let t1 ě t2 ě ¨¨¨ě tr ě 0 be the squares of the nontrivial population canonical correlation coefficients between rx and r y, and let r λ1 ě r λ2 ě ¨¨¨ě r λp^q ě 0 be the squares of the sample canonical correlation coefficients. If the entries of x, y and z are i.i.d. Gaussian, then the following dichotomy has been shown in [7] for a fixed threshold tc P p0, 1q: for any 1 ď i ď r, if ti ă tc, then r λi converges to the right-edge λ`of the limiting eigenvalue spectrum of the sample canonical correlation matrix, and moreover, n 2{3 p r λi ´λ`q converges weakly to the Tracy-Widom law; if ti ą tc, then r λi converges to a deterministic limit θi P pλ`, 1q that is determined by c1, c2 and ti. In this paper, we prove that these results hold universally under the sharp fourth moment conditions on the entries of x, y and z. Moreover, we prove the results in full generality, in the sense that they also hold for near-degenerate ti's and for ti's that are close to the threshold tc. Finally, we also provide almost sharp convergence rates for the sample canonical correlation coefficients under a general a-th moment assumption.

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Singularities of the density of states of random Gram matrices

Cited by 18 publications

References 10 publications

A graphon approach to limiting spectral distributions of Wigner‐type matrices

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

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