Spectral localization in the hierarchical Anderson model

Kritchevski, Evgenij

doi:10.1090/s0002-9939-06-08614-x

Cited by 35 publications

(43 citation statements)

References 8 publications

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“…Due to the use of ergodicity, our Lemmas 2.2 and 3.3 give slightly stronger results, resp. have shorter proofs, than corresponding statements in [Kr1,Kr2,Kr3], who argued without it.…”

Section: Appendix a Ergodicitymentioning

confidence: 87%

Lifshits Tails in the Hierarchical Anderson Model

Kuttruf

Müller

2011

Ann. Henri Poincaré

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Abstract. We prove that the homogeneous hierarchical Anderson model exhibits a Lifshits tail at the upper edge of its spectrum. The Lifshits exponent is given in terms of the spectral dimension of the homogeneous hierarchical structure. Our approach is based on Dirichlet-Neumann bracketing for the hierarchical Laplacian and a large-deviation argument. Mathematics Subject Classification (2010). Primary 47B80; Secondary 81Q10, 93A13 .

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“…Due to the use of ergodicity, our Lemmas 2.2 and 3.3 give slightly stronger results, resp. have shorter proofs, than corresponding statements in [Kr1,Kr2,Kr3], who argued without it.…”

Section: Appendix a Ergodicitymentioning

confidence: 87%

Lifshits Tails in the Hierarchical Anderson Model

Kuttruf

Müller

2011

Ann. Henri Poincaré

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“…Therefore, close to the upper spectral edge E pure ∞ , the integrated DOS exhibits the asymptotic behaviour 8,12,14,15 …”

Section: -23mentioning

confidence: 96%

Renormalization flow of the hierarchical Anderson model at weak disorder

2014

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We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is the only stable fixed-point of the flow equations, whereas for sufficiently slowly decaying hopping energy the renormalized parameters flow to a delta peak fixed-point distribution.In an intermediate range of the hopping decay, both fixed-point distributions are stable and the stationary solution is determined by the initial configuration of the random parameters. We present results for the critical decay of the hopping energy separating the different regimes.

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“…The corresponding hierarchical Laplacian L was studied in a variety of papers, see e.g. [11], [2], [29], [30], [26], [28], [32], [34], [17]. It is the generator of a Markov process, which in the situation of discrete time and space is a random walk on an inifinitely generated group as mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Oscillating heat kernels on ultrametric spaces

2018

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on the occasion of his 60th birthday Streszczenie Let (X, d) be a proper ultrametric space. Given a measure m on X and a function B → C(B) defined on the collection of all nonsingleton balls B of X, we consider the associated hierarchical Laplacian L = L C . The operator L acts in L 2 (X, m), is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel p(t, x, y) with respect to m. We consider the case when X has a transitive group of isometries under which the operator L is invariant and study the asymptotic behaviour of the function t → p(t, x, x) = p(t). It is completely monotone, but does not vary regularly. When X = Q p , the ring of p-adic numbers, and L = D α , the operator of fractional derivative of order α, we show that p(t) = t −1/α A(log p t), where A(τ ) is a continuous non-constant α-periodic function. We also study asymptotic behaviour of min A and max A as the space parameter p tends to ∞. When X = S ∞ , the infinite symmetric group, and L is a hierarchical Laplacian with metric structure analogous to D α , we show that, contrary to the previous case, the completely monotone function p(t) oscillates between two functions ψ(t) and Ψ(t) such that ψ(t)/Ψ(t) → 0 as t → ∞ .

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Spectral localization in the hierarchical Anderson model

Abstract: Abstract. We prove that a large class of hierarchical Anderson models with spectral dimension d ≤ 2 has only pure point spectrum.

Cited by 35 publications

References 8 publications

Lifshits Tails in the Hierarchical Anderson Model

Lifshits Tails in the Hierarchical Anderson Model

Renormalization flow of the hierarchical Anderson model at weak disorder

Oscillating heat kernels on ultrametric spaces

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