Absolutely continuous spectrum for random operators on trees of finite cone type

Keller, Matthias; Lenz, Daniel; Warzel, Simone

doi:10.1007/s11854-012-0040-4

Cited by 31 publications

(54 citation statements)

References 29 publications

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“…A new proof was found in [24], and it was later shown in [6,7] that spectral delocalization and ballistic transport hold in larger regions of the spectrum. Similar results were obtained for more general tree models in [31,25]. One reason that makes trees technically simpler to analyze is the fact that the Green function on a tree satisfies some convenient recursion and factorization relations.…”

Section: Introductionsupporting

confidence: 75%

Quantum ergodicity for the Anderson model on regular graphs

Anantharaman

Sabri

2017

Journal of Mathematical Physics

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We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem

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Section: Introductionsupporting

confidence: 75%

Quantum ergodicity for the Anderson model on regular graphs

Anantharaman

Sabri

2017

Journal of Mathematical Physics

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“…In this case, the spectrum is purely absolutely continuous. In fact, the authors in [18] establish more generally that the absolutely continuous spectrum of A remains stable under small radially symmetric perturbations H = A + λV , if the tree is non-regular. Large parts of the absolutely continuous spectrum also remain stable under small random perturbations H ω = A+λV ω ; see [19].…”

Section: Introductionmentioning

confidence: 99%

Poisson kernel expansions for Schrödinger operators on trees

Anantharaman

Sabri

2018

J. Spectr. Theory

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We study Schrödinger operators on trees and construct associated Poisson kernels, in analogy to the laplacian on the unit disc. We show that in the absolutely continuous spectrum, the generalized eigenfunctions of the operator are generated by the Poisson kernel. We use this to define a "Fourier transform", giving a Fourier inversion formula and a Plancherel formula, where the domain of integration runs over the energy parameter and the geometric boundary of the tree. R

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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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Absolutely continuous spectrum for random operators on trees of finite cone type

Cited by 31 publications

References 29 publications

Quantum ergodicity for the Anderson model on regular graphs

Quantum ergodicity for the Anderson model on regular graphs

Poisson kernel expansions for Schrödinger operators on trees

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

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