I am looking for a good, not too technical discussion of Anderson Localization, and some explanation of why it exists. Googling "Anderson Localization" produces an infinite number of possibilities, so perhaps someone knowledgeable can recommend something...
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asked Sep 10, 2014 at 16:08
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2$\begingroup$ The Aizenman-Molchanov paper made quite a splash when it appeared as it derives the basic results and completely avoids the heavy multi-scale analysis machinery that was previously considered an integral part of the area. (You were probably interested in a survey, but this paper isn't hard to read either.) $\endgroup$– Christian RemlingCommented Sep 10, 2014 at 16:26
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$\begingroup$ @ChristianRemling Thanks! I will check it out! $\endgroup$– Igor RivinCommented Sep 10, 2014 at 16:48
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4 Answers
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For a "canonical" list of references you might consult 50 Years of Anderson Localization. In addition to the Aizenman-Molchanov paper mentioned by Christian Remling, the earlier Fröhlich-Spencer work was also quite influential.
A recent overview of the mathematics of Anderson localization is given by Günter Stolz:
We give a widely self-contained introduction to the mathematical theory of the Anderson model. After defining the Anderson model and determining its almost sure spectrum, we prove localization properties of the model. Here we discuss spectral as well as dynamical localization and provide proofs based on the fractional moments (or Aizenman-Molchanov) method. We also discuss, in less self-contained form, the extension of the fractional moment method to the continuum Anderson model. Finally, we mention major open problems.
We do not aim at the most general known results, but rather want to demonstrate that simple and natural mathematical ideas can be used to rigorously establish Anderson localization.
answered Sep 10, 2014 at 17:40
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1$\begingroup$ The first link has rotted ("The owner of andersonlocalization.com is offering it for sale for an asking price of 620 USD!"). See the proceedings page and the Physics Today article $\endgroup$ Commented Aug 24, 2017 at 15:13
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There is also a new method which gets a hold of eigenvectors directly by an iterative diagonalization procedure rather than indirectly via expectations of products of resolvents. It is in the recent article "Multi-Scale Jacobi Method for Anderson Localization" by John Imbrie.
I should add that the hot topic in the area now is many-body localization with tons of physics articles posted on the cond-mat section of arXiv. The above article by Imbrie can serve as an introduction to his other one about MBL: "On many-body localization for quantum spin chains" in JSP 2016.
answered Sep 12, 2014 at 18:06
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In addition to the literature already recommended, maybe the following references will be also useful:
http://iopscience.iop.org/0034-4885/56/12/001 (Localization: theory and experiment, by B. Kramer and A. MacKinnon)
http://arxiv.org/abs/1005.0915 (Disorder and interference: localization phenomena, by C.A. Müller and D. Delande).
answered Nov 11, 2014 at 13:12
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Not technical, and not a complete answer, but illuminating:
https://www.quantamagazine.org/mathematicians-tame-rogue-waves-illuminating-future-of-led-lighting-20170822/
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2$\begingroup$ The referenced papers for that article: arxiv.org/pdf/1107.0397.pdf and arxiv.org/pdf/1704.05512.pdf $\endgroup$– AHusainCommented Aug 25, 2017 at 2:30