Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

Helton, J. William; Far, Reza Rashidi; Speicher, Roland

doi:10.1093/imrn/rnm086

Cited by 81 publications

(144 citation statements)

References 11 publications

(12 reference statements)

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“…The choice of the correct metric on this space follows naturally from the general theory by Earle and Hamilton [17]. The same line of reasoning has appeared before in a context close to ours in [21,26,27].…”

Section: A Existence and Uniquenessmentioning

confidence: 55%

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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Section: A Existence and Uniquenessmentioning

confidence: 55%

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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“…The question of existence and uniqueness of solutions to (2.2) with the constraint (2.3) has been answered in [38]. The MDE has a unique solution matrix M(ζ ) for any spectral parameter ζ ∈ H and these matrices constitute a holomorphic function M : H → C N ×N .…”

Section: Note That In Particular S Commutes With Taking the Adjoint mentioning

confidence: 99%

“…Let H = (h x,y ) N x,y=1 ∈ C N ×N be a self-adjoint random matrix. For a spectral parameter ζ ∈ H we consider the associated Matrix Dyson Equation (2.20) preserves the cone C + of positive semidefinite matrices and the MDE therefore has a unique solution [38] whose properties have been presented in Sect. 2.1.…”

Section: Random Matrices With Correlationsmentioning

confidence: 99%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

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“…In [12], this convolution is analyzed in great detail: among others, a formula for the density of the corresponding distribution is provided, and it is shown that this density is bounded, continuous and analytic wherever positive. However, in order to study the asymptotic eigenvalue distribution of more general selfadjoint polynomials P (A N , H N ) it is necessary to consider the more general framework of free convolutions of operator-valued distributions [27,22,23,17,5]. In the present note, we find certain operator-valued counterparts of Biane's results from [12]; necessarily, several of the conclusions of [12] do not hold in this more general setting.…”

Section: Introductionmentioning

confidence: 69%

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

Belinschi

2017

Complex Anal. Oper. Theory

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Abstract. In his article On the free convolution with a semicircular distribution, Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy-Stieltjes transform of the free additive convolution of a probability measure on R with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane's methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars. In addition, it investigates properties of the Julia-Carathéodory derivative of the subordination functions associated to such semigroups, extending results from [7].

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Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

Cited by 81 publications

References 11 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

Stability of the matrix Dyson equation and random matrices with correlations

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

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