Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges

Germinet, François; Klopp, Frédéric

doi:10.1007/s00023-012-0217-5

Cited by 26 publications

(23 citation statements)

References 27 publications

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“…In Section 6, we review and comment results on the asymptotic expansion formulas and Poisson limit theorems for the largest eigenvalues λ K,V as well as localization properties of the corresponding eigenfunctions. These results are proved by Astrauskas and Molchanov (1992), Gärtner and Molchanov (1998), Astrauskas (2007;2012;2013), Germinet and Klopp (2013), Biskup and König (2016) and other mathematicians. These papers are complemented by the present survey on the asymptotic geometric properties of ξ V -extremes and related RV classes of distributions.…”

Section: Extreme Value Theory For Eigenvalues Of Large-volume Andersomentioning

confidence: 71%

“…To the best of our knowledge, Poisson limit theorems for the (unfolded) largest eigenvalues were proved only in the case ν = 1 and bounded ξ(0), provided the distribution 1− e −Q satisfies additional continuity and tail decay conditions (Germinet and Klopp 2013); see also Section 6.2 below. For ν 2 or general RV conditions on Q satisfying (1.13), the Poissonian convergence of the top eigenvalues still remains an open problem.…”

Section: Extreme Value Theory For Eigenvalues Of Large-volume Andersomentioning

confidence: 99%

“…Recently, Germinet and Klopp (2013;) presented a comprehensive study on limit theorems for eigenvalues of large-volume Hamiltonians in the spectral regions I pp of Anderson localization. For fixed λ 0 ∈ I pp as above, the authors considered the unfolded eigenvalues…”

Section: Relations To Infinite-volume Anderson Hamiltoniansmentioning

confidence: 99%

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From extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonian

Astrauskas¹

2016

Probab. Surveys

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The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which unboundedly increases. We discuss the following topics: (I) high level exceedances, in particular, clustering of exceedances; (II) decay rate of spacings in comparison with increasing rate of extreme order statistics; (III) minimum of spacings of successive order statistics; (IV) asymptotic behavior of values neighboring to extremes and so on. The conditions of the results are formulated in terms of regular variation (RV) of the cumulative hazard function and its inverse. A relationship between RV classes of the present paper as well as their links to the well-known RV classes (including domains of attraction of max-stable distributions) are discussed.The asymptotic behavior of functionals (I)-(IV) determines the asymptotic structure of the top eigenvalues and the corresponding eigenfunctions of the large-volume discrete Schrödinger operators with an i.i.d. potential (Anderson Hamiltonian). Thus, another aim of the present paper is to review and comment a recent progress on the extreme value theory for eigenvalues of random Schrödinger operators as well as to provide a clear and rigorous understanding of the relationship between the top eigenvalues and extreme values of i.i.d. random potentials. We also discuss their links to the long-time intermittent behavior of the parabolic problems associated with the Anderson Hamiltonian via spectral representation of solutions.

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Section: Extreme Value Theory For Eigenvalues Of Large-volume Andersomentioning

confidence: 71%

Section: Extreme Value Theory For Eigenvalues Of Large-volume Andersomentioning

confidence: 99%

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From extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonian

Astrauskas¹

2016

Probab. Surveys

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“…The first result on the convergence of point processes of both the eigenvalues and the concentration centres of the eigenfunctions is [KilNak07]; see also [Nak07]. The currently strongest available results are in [GerKlo14] and [GerKlo13], where [GerKlo14] works in the bulk of the spectrum and [GerKlo13] close to the top; see also [GerKlo11].…”

Section: Relation To Anderson Localisationmentioning

confidence: 99%

“…Since 0 is assumed to lie in the interior of the support of the IDS, also [GerKlo13] makes assertions only for eigenvalues that are substantially away from the boundary of the spectrum (however, it contains also a restricted assertion precisely at the boundary for the one-dimensional operator H D d C ). k .B L / a L /b L for box-depending quantities a L and b L , but on the unfolded eigenvalues OE .…”

Section: Relation To Anderson Localisationmentioning

confidence: 99%

The Parabolic Anderson Model

König

2016

Pathways in Mathematics

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Each "Pathways in Mathematics" book offers a roadmap to a currently well developing mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educational style, i.e., in a way that is accessible for advanced undergraduate and graduate students. It also serves as an introduction to and survey of the field for researchers who want to be quickly informed about the state of the art. The point of departure is typically a bachelor/masters level background, from which the reader is expeditiously guided to the frontiers. This is achieved by focusing on ideas and concepts underlying the development of the subject while keeping technicalities to a minimum. Each volume contains an extensive annotated bibliography as well as a discussion of open problems and future research directions as recommendations for starting new projects More information about this series at

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Interacting Electrons in a Random Medium: A Simple One-Dimensional Model

Klopp

Veniaminov

2020

Frontiers in Analysis and Probability

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The present paper is devoted to the study of a simple model of interacting electrons in a random background. In a large interval Λ, we consider n one dimensional particles whose evolution is driven by the Luttinger-Sy model, i.e., the interval Λ is split into pieces delimited by the points of a Poisson process of intensity µ and, in each piece, the Hamiltonian is the Dirichlet Laplacian. The particles interact through a repulsive pair potential decaying polynomially fast at infinity. We assume that the particles have a positive density, i.e., n/|Λ| → ρ > 0 as |Λ| → +∞. In the low density or large disorder regime, i.e., ρ/µ small, we obtain a two term asymptotic for the thermodynamic limit of the ground state energy per particle of the interacting system; the first order correction term to the non interacting ground state energy per particle is controlled by pairs of particles living in the same piece. The ground state is described in terms of its one and two-particles reduced density matrix. Comparing the interacting and the non interacting ground states, one sees that the effect of the repulsive interactions is to move a certain number of particles living together with another particle in a single piece to a new piece that was free of particles in the non interacting ground state. 1. Introduction: the model and the main results On R, consider a Poisson point process dµ(ω) of intensity µ. Let (x k (ω)) k∈Z denote its support (i.e., dµ(ω) = k∈Z δ x k (ω)), the points being ordered increasingly.

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Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges

Cited by 26 publications

References 27 publications

From extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonian

From extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonian

The Parabolic Anderson Model

Interacting Electrons in a Random Medium: A Simple One-Dimensional Model

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