NO EIGENVALUES OUTSIDE THE SUPPORT OF THE LIMITING SPECTRAL DISTRIBUTION OF INFORMATION-PLUS-NOISE TYPE MATRICES

2012 ◽  
Vol 01 (01) ◽  
pp. 1150004 ◽  
Author(s):  
ZHIDONG BAI ◽  
JACK W. SILVERSTEIN
Keyword(s):  
Probability Distribution ◽  
Spectral Distribution ◽  
Correlation Matrix ◽  
Closed Interval ◽  
Empirical Distribution ◽  
Limiting Distribution ◽  
Limiting Spectral Distribution ◽  
Noise Type ◽  
Sample Correlation ◽  
Class Of Matrices

We consider a class of matrices of the form Cn = (1/N)(Rn + σXn)(Rn + σXn)*, where Xn is an n × N matrix consisting of independent standardized complex entries, Rn is an n × N nonrandom matrix, and σ > 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in [Formula: see text], but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of [Formula: see text] converge to a proper probability distribution, then the empirical distribution of the eigenvalues of Cn converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on Rn, for any closed interval in ℝ+ outside the support of the limiting distribution, then, almost surely, no eigenvalues of Cn will appear in this interval for all n large.

scholarly journals On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Ion Nechita
Keyword(s):  
Probability Distribution ◽  
Spectral Distribution ◽  
Quantum States ◽  
The Other ◽  
Unitary Operators ◽  
Matrix Size ◽  
Limiting Spectral Distribution ◽  
Large Matrix ◽  
Random Quantum States ◽  
Invariant Quantum

We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.

Some strong convergence theorems for eigenvalues of general sample covariance matrices

2021 ◽  
Author(s):  
Yanqing Yin
Keyword(s):  
General Population ◽  
Strong Convergence ◽  
Spectral Properties ◽  
Closed Interval ◽  
Open Interval ◽  
Convergence Result ◽  
Covariance Matrices ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Class Of Matrices

The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form [Formula: see text], where [Formula: see text] is a [Formula: see text] nonrandom matrix and [Formula: see text] is an [Formula: see text] matrix consisting of i.i.d standard complex entries. [Formula: see text] as [Formula: see text] while [Formula: see text] can be arbitrary but no smaller than [Formula: see text]. We first prove that under some mild assumptions, with probability 1, for all large [Formula: see text], there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large [Formula: see text]. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.

scholarly journals Brown measure and asymptotic freeness of elliptic and related matrices

2019 ◽  
Vol 08 (02) ◽  
pp. 1950007
Author(s):  
Kartick Adhikari ◽  
Arup Bose
Keyword(s):  
Spectral Distribution ◽  
Limiting Spectral Distribution

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.

scholarly journals On the spectral distribution of large weighted random regular graphs

2014 ◽  
Vol 03 (04) ◽  
pp. 1450015 ◽  
Author(s):  
Leo Goldmakher ◽  
Cap Khoury ◽  
Steven J. Miller ◽  
Kesinee Ninsuwan
Keyword(s):  
Spectral Distribution ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Spectral Measures ◽  
Limiting Spectral Distribution ◽  
Random Weights ◽  
Large Trees ◽  
The Difference ◽  
The Relationship ◽  
Limiting Spectral Measure

McKay proved the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. Given a large d-regular graph we assign random weights, drawn from some distribution [Formula: see text], to its edges. We study the relationship between [Formula: see text] and the associated limiting spectral distribution obtained by averaging over the weighted graphs. We establish the existence of a unique "eigendistribution" (a weight distribution [Formula: see text] such that the associated limiting spectral distribution is a rescaling of [Formula: see text]). Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d2)). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

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