A CLT for a band matrix model

Anderson, Greg W.; Zeitouni, Ofer

doi:10.1007/s00440-004-0422-3

Cited by 178 publications

(339 citation statements)

References 25 publications

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“…The following result is considered folklore in the literature (see e.g. [8,23,26,27,28]). For completeness we include its proof, adjusted to our setup, in the Appendix A.…”

Section: Resultsmentioning

confidence: 99%

“…For a proof of (3.1) see [24,37] (in the Gaussian setting), [8] (with the additional condition that the fourth moments have a profile), and [5] (with bounded higher moments). Bounded moment conditions can be relaxed using a standard cut-off argument (cf.…”

Section: Wigner Type Matricesmentioning

confidence: 99%

“…In particular, if (1.1) is stable under small perturbations, then g ii is close to m i and the average N −1 ∑ i m i approximates the normalized trace of the resolvent, N −1 Tr G. Being determined by N −1 Im Tr G, as Im z → 0, the empirical spectral measure of H approaches the non-random measure with density 3) as N goes to infinity, see [8,24,37].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Ajanki

Krüger

Erdős

2016

Comm Pure Appl Math

105

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Let S be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation − 1 m = z + Sm with a parameter z in the complex upper half-plane H has a unique solution m with values in H. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on R. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most three.Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles; (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or a cubic root cusps; no other singularities occur.

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“…The following result is considered folklore in the literature (see e.g. [8,23,26,27,28]). For completeness we include its proof, adjusted to our setup, in the Appendix A.…”

Section: Resultsmentioning

confidence: 99%

Section: Wigner Type Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Ajanki

Krüger

Erdős

2016

Comm Pure Appl Math

105

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show abstract

“…Much of this attention has focused on scenarios with identity population covariance (e.g., [CL98,LP09,AZ05]), although some results for more general matrix models have also appeared [BS04].…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models

Passemier

McKay

Chen

2015

Journal of Multivariate Analysis

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Abstract. This paper derives central limit theorems (CLTs) for general linear spectral statistics (LSS) of three important multi-spiked Hermitian random matrix ensembles. The first is the most common spiked scenario, proposed by Johnstone, which is a central Wishart ensemble with fixed-rank perturbation of the identity matrix, the second is a non-central Wishart ensemble with fixed-rank noncentrality parameter, and the third is a similarly defined non-central F ensemble. These CLT results generalize our recent work [PMC14] to account for multiple spikes, which is the most common scenario met in practice. The generalization is non-trivial, as it now requires dealing with hypergeometric functions of matrix arguments. To facilitate our analysis, for a broad class of such functions, we first generalize a recent result of Onatski [Ona14] to present new contour integral representations, which are particularly suitable for computing large-dimensional properties of spiked matrix ensembles. Armed with such representations, our CLT formulas are derived for each of the three spiked models of interest by employing the Coulomb fluid method from random matrix theory along with saddlepoint techniques. We find that for each matrix model, and for general LSS, the individual spikes contribute additively to yield a O(1) correction term to the asymptotic mean of the linear statistic, which we specify explicitly, whilst having no effect on the leading order terms of the mean or variance.

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“…Condition (1) implies that B u,v vanishes when uv is not an edge and that for small ε the entries of B + εM are in [0, 1]. For a finite G, let μ ε denote the empirical probability measure of the eigenvalues…”

Section: Introductionmentioning

confidence: 99%

The spectrum of the random environment and localization of noise

Cheliotis

Virág

2009

Probab. Theory Relat. Fields

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We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.

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A CLT for a band matrix model

Cited by 178 publications

References 25 publications

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models

The spectrum of the random environment and localization of noise

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