In this Review Article, we review the results of Anderson localization for different random families of operators that enter the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical points of view. From the Anderson–Bernoulli conjecture in dimension 2, we justify the introduction of quasi-one-dimensional models. Then, we present different types of these models: the Schrödinger type in the discrete and continuous cases, the unitary type, the Dirac type, and the point interaction type. We present tools coming from the study of dynamical systems in dimension one: the transfer matrix formalism, the Lyapunov exponents, and the Furstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schrödinger type involving only geometric and algebraic properties of the Furstenberg group. Then, we review results of localization, first for Schrödinger-type models and then for unitary type models. Each time, we reduce the question of localization to the study of the Furstenberg group and show how to use more and more refined algebraic criteria to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schrödinger type include the case of Bernoulli randomness.
REFERENCES
1.
Ahlbrecht
, A.
, Scholz
, V. B.
, and Werner
, A. H.
, “Disordered quantum walks in one lattice dimension
,” J. Math. Phys.
52
, 102201
(2011
).2.
Aizenman
, M.
and Warzel
, S.
, Random Operators: Disorder Effects on Quantum Spectra and Dynamics
[American Mathematical Society (AMS)
, Providence, RI
, 2015
], Vol. 168
, pp. xiv+326
.3.
Albeverio
, S.
, Gesztesy
, F.
, Høegh-Krohn
, R.
, and Holden
, H.
, Solvable Models in Quantum Mechanics. With an Appendix by Pavel Exner
(AMS Chelsea Publishing
, Providence, RI
, 2005
), pp. xiv+488
.4.
Anderson
, P.
, Basic Notions of Condensed Matter Physics
(CRC Press
, 2018
).5.
Anderson
, P. W.
, “Absence of diffusion in certain random lattices
,” Phys. Rev.
109
, 1492
–1505
(1958
).6.
Arnold
, L.
, Random Dynamical Systems
(Springer
, Berlin
, 1998
), pp. xi+586
.7.
Asch
, J.
, Bourget
, O.
, and Joye
, A.
, “Localization properties of the Chalker–Coddington model
,” Ann. Henri Poincaré
11
, 1341
–1373
(2010
).8.
Asch
, J.
, Bourget
, O.
, and Joye
, A.
, “Dynamical localization of the Chalker-Coddington model far from transition
,” J. Stat. Phys.
147
, 194
–205
(2012
).9.
Basko
, D. M.
, Aleiner
, I. L.
, and Altshuler
, B. L.
, “Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states
,” Ann. Phys.
321
, 1126
–1205
(2006
).10.
Berezin
, F. A.
and Faddeev
, L. D.
, “A remark on Schrödinger’s equation with a singular potential
,” Sov. Math., Dokl.
2
, 372
–375
(1961
).11.
Bougerol
, P.
and Lacroix
, J.
, Products of Random Matrices with Applications to Schrödinger Operators
(Birkhäuser
, Boston, MA
, 1985
), Vol. 8
.12.
Boumaza
, H.
, “Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model
,” Math. Phys., Anal. Geom.
10
, 97
–122
(2007
).13.
Boumaza
, H.
, “Hölder continuity of the integrated density of states for matrix-valued Anderson models
,” Rev. Math. Phys.
20
, 873
–900
(2008
).14.
Boumaza
, H.
, “A matrix-valued point interactions model
,” Lett. Math. Phys.
87
, 81
–97
(2009
).15.
Boumaza
, H.
, “Localization for a matrix-valued Anderson model
,” Math. Phys., Anal. Geom.
12
, 255
–286
(2009
).16.
Boumaza
, H.
, “Localization for an Anderson-Bernoulli model with generic interaction potential
,” Tohoku Math. J.
65
, 57
–74
(2013
).17.
Boumaza
, H.
and Marin
, L.
, “Absence of absolutely continuous spectrum for random scattering zippers
,” J. Math. Phys.
56
, 022701
(2015
).18.
Boumaza
, H.
and Najar
, H.
, “Lifshitz tails for continuous matrix-valued Anderson models
,” J. Stat. Phys.
160
, 371
–396
(2015
).19.
Boumaza
, H.
and Stolz
, G.
, “Positivity of Lyapunov exponents for Anderson-type models on two coupled strings
,” Electron. J. Differ. Equations
18
, 47
(2007
).20.
Bourgain
, J.
and Kenig
, C. E.
, “On localization in the continuous Anderson-Bernoulli model in higher dimension
,” Invent. Math.
161
, 389
–426
(2005
).21.
Anderson Localization and Its Ramification. Disorder, Phase Coherence, and Electron Correlations
, Lecture Notes in Physics Vol. 630
, edited by Brandes
, T.
and Kettemann
, S.
(Springer
, Berlin
, 2003
).22.
Breuillard
, E.
, Cours Peccot 2006: Propriétés qualitatives des groupes discrets, 2006
.23.
Breuillard
, E.
and Gelander
, T.
, “On dense free subgroups of Lie groups
,” J. Algebra
261
, 448
–467
(2003
).24.
Bucaj
, V.
, Damanik
, D.
, Fillman
, J.
, Gerbuz
, V.
, Vandenboom
, T.
, Wang
, F.
, and Zhang
, Z.
, “Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
,” Trans. Am. Math. Soc.
372
, 3619
–3667
(2019
).25.
Bucaj
, V.
, Damanik
, D.
, Fillman
, J.
, Gerbuz
, V.
, VandenBoom
, T.
, Wang
, F.
, and Zhang
, Z.
, “Positive Lyapunov exponents and a large deviation theorem for continuum Anderson models, briefly
,” J. Funct. Anal.
277
, 3179
–3186
(2019
).26.
Cantero
, M. J.
, Grünbaum
, F. A.
, Moral
, L.
, and Velázquez
, L.
, “One-dimensional quantum walks with one defect
,” Rev. Math. Phys.
24
, 1250002
(2012
).27.
Cantero
, M. J.
, Moral
, L.
, Grünbaum
, F. A.
, and Velázquez
, L.
, “Matrix-valued Szegő polynomials and quantum random walks
,” Commun. Pure Appl. Math.
63
, 464
–507
(2010
).28.
Cantero
, M. J.
, Moral
, L.
, and Velázquez
, L.
, “Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle
,” Linear Algebra Appl.
362
, 29
–56
(2003
).29.
Carmona
, R.
, Klein
, A.
, and Martinelli
, F.
, “Anderson localization for Bernoulli and other singular potentials
,” Commun. Math. Phys.
108
, 41
–66
(1987
).30.
Carmona
, R.
and Lacroix
, J.
, Spectral Theory of Random Schrödinger Operators
(Birkhäuser Verlag
, Basel
, 1990
), pp. xxvi+587
.31.
Chapman
, J.
and Stolz
, G.
, “Localization for random block operators related to the XY spin chain
,” Ann. Henri Poincaré
16
, 405
–435
(2015
).32.
Damanik
, D.
, Fillman
, J.
, Helman
, M.
, Kesten
, J.
, and Sukhtaiev
, S.
, “Random Hamiltonians with arbitrary point interactions in one dimension
,” J. Differ. Equations
282
, 104
–126
(2021
).33.
Damanik
, D.
, Fillman
, J.
, and Sukhtaiev
, S.
, “Localization for Anderson models on metric and discrete tree graphs
,” Math. Ann.
376
, 1337
–1393
(2020
).34.
Damanik
, D.
, Sims
, R.
, and Stolz
, G.
, “Localization for one-dimensional, continuum, Bernoulli-Anderson models
,” Duke Math. J.
114
, 59
–100
(2002
).35.
Damanik
, D.
and Stollmann
, P.
, “Multi-scale analysis implies strong dynamical localization
,” Geom. Funct. Anal.
11
, 11
–29
(2001
).36.
Delyon
, F.
, Simon
, B.
, and Souillard
, B.
, “From power pure point to continuous spectrum in disordered systems
,” Ann. I.H.P.: Phys. Theor.
42
, 283
–309
(1985
).37.
Ding
, J.
and Smart
, C. K.
, “Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
,” Invent. Math.
219
, 467
–506
(2020
).38.
Disertori
, M.
, Kirsch
, W.
, Klein
, A.
, Klopp
, F.
, and Rivasseau
, V.
, Random Schrödinger Operators
[Société Mathématique de France (SMF)
, Paris
, 2008
], Vol. 25
, pp. xiv+213
.39.
Duarte
, P.
and Klein
, S.
, “Large deviations for products of random two dimensional matrices
,” Commun. Math. Phys.
375
, 2191
–2257
(2020
).40.
Furstenberg
, H.
, “Noncommuting random products
,” Trans. Am. Math. Soc.
108
, 377
–428
(1963
).41.
Furstenberg
, H.
and Kesten
, H.
, “Products of random matrices
,” Ann. Math. Stat.
31
, 457
–469
(1960
).42.
Ge
, L.
and Zhao
, X.
, “Exponential dynamical localization in expectation for the one dimensional Anderson model
,” J. Spectral Theory
10
, 887
–904
(2020
).43.
Genovese
, G.
, Giacomin
, G.
, and Greenblatt
, R. L.
, “Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices
,” Commun. Math. Phys.
351
, 923
–958
(2017
).44.
Germinet
, F.
and Klein
, A.
, “Bootstrap multiscale analysis and localization in random media
,” Commun. Math. Phys.
222
, 415
–448
(2001
).45.
Germinet
, F.
, Klein
, A.
, and Schenker
, J. H.
, “Dynamical delocalization in random Landau Hamiltonians
,” Ann. Math.
166
(1
), 215
–244
(2007
).46.
Girvin
, S.
and Yang
, K.
, Modern Condensed Matter Physics
(Cambridge University Press
, 2019
).47.
Glaffig
, C.
, “Smoothness of the integrated density of states on strips
,” J. Funct. Anal.
92
, 509
–534
(1990
).48.
Goldsheid
, I. Y.
, “Zariski closure of subgroups of the symplectic group and Lyapunov exponents of the Schrödinger operator on the strip
,” Commun. Math. Phys.
174
, 347
–365
(1995
).49.
Goldsheid
, I. Y.
and Margulis
, G. A.
, “Lyapunov indices of a product of random matrices
,” Russ. Math. Surv.
44
, 11
–71
(1989
).50.
Gorodetski
, A.
and Kleptsyn
, V.
, “Parametric Furstenberg theorem on random products of matrices
,” Adv. Math.
378
, 107522
(2021
).51.
Guivarc’h
, Y.
and Raugi
, A.
, “Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence
,” Z. Wahrscheinlichkeitstheorie Verw. Geb.
69
, 187
–242
(1985
).52.
Hamza
, E.
, Joye
, A.
, and Stolz
, G.
, “Dynamical localization for unitary Anderson models
,” Math. Phys., Anal. Geom.
12
, 381
–444
(2009
).53.
Hamza
, E.
and Stolz
, G.
, “Lyapunov exponents for unitary Anderson models
,” J. Math. Phys.
48
, 043301
(2007
).54.
Hislop
, P. D.
, Fourth Summer School in Analysis and Mathematical Physics: Topics in Spectral Theory and Quantum Mechanics, Cuernavaca, México, May 2005
, Lectures on Random Schrödinger Operators
[American Mathematical Society (AMS)
, Providence, RI
, 2008
], pp. 41
–131
.55.
Hislop
, P. D.
, Kirsch
, W.
, and Krishna
, M.
, “Spectral and dynamical properties of random models with nonlocal and singular interactions
,” Math. Nachr.
278
, 627
–664
(2005
).56.
Hislop
, P. D.
, Kirsch
, W.
, and Krishna
, M.
, “Eigenvalue statistics for Schrödinger operators with random point interactions on
,” J. Math. Phys.
61
, 092103
(2020
).57.
Hurt
, N. E.
, Mathematical Physics of Quantum Wires and Devices: From Spectral Resonances to Anderson Localization
, Mathematics and its Applications Vol. 506
(Kluwer Academic Publishers
, Dordrecht
, 2000
).58.
Jitomirskaya
, S.
and Zhu
, X.
, “Large deviations of the Lyapunov exponent and localization for the 1D Anderson model
,” Commun. Math. Phys.
370
, 311
–324
(2019
).59.
Kingman
, J. F. C.
, “Subadditive ergodic theory
,” Ann. Probab.
1
, 883
–909
(1973
).60.
Kirsch
, W.
and Martinelli
, F.
, “On the ergodic properties of the spectrum of general random operators
,” J. Reine Angew. Math.
334
, 141
–156
(1982
).61.
Klein
, A.
, “Multiscale analysis and localization of random operators
,” in Random Schrödinger Operators
[Société Mathématique de France (SMF)
, Paris
, 2008
], pp. 221
–259
.62.
Klein
, A.
, Lacroix
, J.
, and Speis
, A.
, “Localization for the Anderson model on a strip with singular potentials
,” J. Funct. Anal.
94
, 135
–155
(1990
).63.
Kotani
, S.
, “Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators
,” in Stochastic Analysis: Proceedings of the Taniguchi International Symposium on Stochastic Analysis, Katata and Kyoto, 1982
, North-Holland Mathematical Library Vol. 32
(Elsevier
, 1984
), pp. 225
–247
.64.
Kotani
, S.
and Simon
, B.
, “Stochastic Schrödinger operators and Jacobi matrices on the strip
,” Commun. Math. Phys.
119
, 403
–429
(1988
).65.
Kunz
, H.
and Souillard
, B.
, “Sur le spectre des opérateurs aux différences finies aléatoires
,” Commun. Math. Phys.
78
, 201
–246
(1980
).66.
Li
, L.
and Zhang
, L.
, “Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle
,” Duke Math. J.
171
, 327
–415
(2022
).67.
Macera
, D.
and Sodin
, S.
, “Anderson localisation for quasi-one-dimensional random operators
,” Ann. Henri Poincaré
23
, 4227
–4247
(2022
).68.
Marin
, L.
and Schulz-Baldes
, H.
, “Scattering zippers and their spectral theory
,” J. Spectral Theory
3
, 47
–82
(2013
).69.
Oseledets
, V. I.
, “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems
,” Trans. Moscow Math. Soc.
19
, 197
–231
(1968
).70.
Pastur
, L. A.
, “Spectral properties of disordered systems in the one-body approximation
,” Commun. Math. Phys.
75
, 179
–196
(1980
).71.
Rangamani
, N.
, “Singular-unbounded random Jacobi matrices
,” J. Math. Phys.
60
, 081904
(2019
).72.
Rangamani
, N.
, “Exponential dynamical localization for random word models
,” Ann. Henri Poincaré
23
, 4171
–4193
(2022
).73.
Reed
, M.
and Simon
, B.
, Methods of Modern Mathematical Physics. IV: Analysis of Operators
(Academic Press
, New York; San Francisco; London
, 1978
), Vol. 15
, p. 396
.74.
Reed
, M.
and Simon
, B.
, Methods of Modern Mathematical Physics. III: Scattering Theory
(Academic Press
, New York; San Francisco; London
, 1979
), Vol. 15
, p. 463
.75.
Ruelle
, D.
, “Ergodic theory of differentiable dynamical systems
,” Publ. Math. l’IHÉS
50
, 27
–58
(1979
).76.
Sadel
, C.
and Schulz-Baldes
, H.
, “Random Dirac operators with time reversal symmetry
,” Commun. Math. Phys.
295
, 209
–242
(2010
).77.
Shubin
, C.
, Vakilian
, R.
, and Wolff
, T.
, “Some harmonic analysis questions suggested by Anderson-Bernoulli models
,” Geom. Funct. Anal.
8
, 932
–964
(1998
), Appendix by T. H. Wolff.78.
Simon
, B.
, Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory
, Colloquium Publications Vol. 54
[American Mathematical Society (AMS)
, Providence, RI
, 2005
].79.
Simon
, B.
, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory
, Colloquium Publications Vol. 54
(American Mathematical Society
, Providence, RI
, 2005
).80.
Stollmann
, P.
, Caught by Disorder
, Bound States in Random Media Vol. 20
(Birkhäuser
, Boston
, 2001
), p. xvi+166
.81.
Veselić
, I.
, Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators
, Lecture Notes in Mathematics Vol. 1917
(Springer
, Berlin
, 2008
).82.
Werner
, A. H.
, “Localization and recurrence in quantum walks
,” Ph.D. thesis, Gottfried Wilhelm Leibniz Universität Hannover
, 2013
.83.
Zalczer
, S.
, “Localization for one-dimensional Anderson–Dirac models
,” Ann. Henri Poincaré
24
, 37
–72
(2023
).84.
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