An introduction to the mathematics of Anderson localization

Stolz, Günter

doi:10.1090/conm/552/10911

Cited by 68 publications

(99 citation statements)

References 34 publications

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“…For d ≥ 2 localization was proved using the estimates of [FS2] for strong disorder λ 1 or for small λ > 0 and E ≤ −Cλ 2 . See [AM] for an elegant proof using fractional moments and [Sp1,Sto] for mathematical reviews of Anderson localization.…”

Section: The Anderson Model and Random Band Matricesmentioning

confidence: 99%

Duality, statistical mechanics and random matrices

Spencer

2012

Current Developments in Mathematics

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Abstract. This article will present an informal review of some results and conjectures about the spectral theory of large random matrices and related spin systems in statistical mechanics. A class of lattice spin models provides a dual representation for spectral problems in random matrix theory. Ordered and disordered phases of the spins correspond to different spectral types and quantum time evolutions. In three dimensions, we describe a phase transition for a supersymmetric statistical mechanics system inspired by random matrix theory. This transition has a classical interpretation in terms of a history dependent walk on the lattice. In the ordered phase the walk is diffusive while in the disordered phase it is localized near its starting point.

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Section: The Anderson Model and Random Band Matricesmentioning

confidence: 99%

Duality, statistical mechanics and random matrices

Spencer

2012

Current Developments in Mathematics

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“…Mathematical study of Anderson localization is an active field; we refer readers to the recent reviews [14,20] on the subject for the detailed bibliography. In this work, we focus our attention on a single aspect of Anderson localizationthe so-called m-level Wegner estimate.…”

Section: Previous Resultsmentioning

confidence: 99%

Eigenvalue counting inequalities, with applications to Schrödinger operators

Elgart¹,

Schmidt²

2015

J. Spectr. Theory

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ABSTRACT. We derive a sufficient condition for a Hermitian N × N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (−ǫ, ǫ). This condition is expressed in terms of the existence of a principal (N − 2m) × (N − 2m) submatrix of A whose Schur complement in A has at least m eigenvalues in the interval (−Kǫ, Kǫ), with an explicit constant K.We apply this result to a random Schrödinger operator H ω , obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for H ω -a result known as an m-level Wegner estimate. We demonstrate its usefulness by verifying the input condition of our criterion for some physical models. These include the Anderson model and random block operators that arise in the Bogoliubov-de Gennes theory of dirty superconductors.

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“…As a result, many of these topics have appeared now for the first time in book form (while partial accounts of the FMM have been provided in other books or book chapters; e.g., [9,19,22]). Given that the fractional moments method provides a very natural and direct entry into understanding Anderson localization, this book has been long awaited and will undoubtedly become an invaluable resource for both researchers and students.…”

Section: Book Reviewsmentioning

confidence: 99%