Absolutely Continuous Spectrum for Random Schrödinger Operators on Tree-Strips of Finite Cone Type

Sadel, Christian

doi:10.1007/s00023-012-0203-y

Cited by 12 publications

(20 citation statements)

References 40 publications

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“…indicates that x is closer to the root of Γ than y. In this example (s xy ) is not a symmetric matrix, but in a related model it may be chosen symmetric (rooted trees of finite cone types associated with a substitution matrix S, see [Sad12]). In particular, real analyticity of the density of states (away from the spectral edges) in this model follows from our analysis (We thank C. Sadel for pointing out this connection).…”

Section: Introductionmentioning

confidence: 99%

Quadratic Vector Equations On Complex Upper Half-Plane

Erdős²,

2019

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We consider the nonlinear equation − 1 m = z + Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In [?] we qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16c], we present a complete stability analysis of the equation for any z ∈ H, including the vicinity of the singularities.

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Section: Introductionmentioning

confidence: 99%

Quadratic Vector Equations On Complex Upper Half-Plane

Erdős²,

2019

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“…For small disorder in the bulk of the spectrum, localization and Poisson statistics appears in one and quasione dimensional systems [GMP, KuS, CKM, Lac, KLS] (except if prevented by a symmetry [SS3]) and is expected (but not proved) in 2 dimensions. Delocalization for the Anderson model was first rigorously proved on regular trees (Bethe lattices) [Kl] and had been extended to several infinitedimensional tree-like graphs [Kl,ASW,FHS,AW,KLW,FHH,KS,Sa2,Sa3,Sha]. Recently it was shown that there is a transition from localization to delocalization on normalized antitrees at exactly 2-dimensional growth rate [Sa4].…”

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confidence: 99%

A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes

Sadel¹,

Virág²

2016

Commun. Math. Phys.

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Abstract:We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix T 0 . Focusing on the eigenvalues of T 0 of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required T 0 to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.

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“…potential) the existence of absolutely continuous spectrum at low disorder has only been shown for trees and treelike graphs 1 of infinite dimension with exponentially growing boundary. A lot of work has been done in extending Klein's original result, also in recent years, [ASW,AW1,AW2,FHS1,FHS2,FHS3,FHH,Ha,KLW,KlS,Sa,Sh]. At large disorder and on the edge of the spectrum one typically finds Anderson localization (pure point spectrum) in any dimension [A, AM, CKM, DK, DLS, FMSS, FS, Klo, W].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Localization for transversally periodic random potentials on binary trees

Froese¹,

Lee²,

Sadel³

et al. 2016

J. Spectr. Theory

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Abstract. We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, Qr, and a random transversally periodic potential, κQt, with coupling constant κ. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large κ. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing κ. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder.

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Absolutely Continuous Spectrum for Random Schrödinger Operators on Tree-Strips of Finite Cone Type

Cited by 12 publications

References 40 publications

Quadratic Vector Equations On Complex Upper Half-Plane

Quadratic Vector Equations On Complex Upper Half-Plane

A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes

Localization for transversally periodic random potentials on binary trees

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