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Random Matrix Theory and the Anderson Model

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Abstract

This paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal-insulator Anderson transition for the Anderson model in dimension d≥3. The possible criterions used to define such a transition are presented. It is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model. Various simplified versions for which rigorous results have been obtained in the past are discussed. It includes a free probability approach, the Wegner n-orbital model and a class of models proposed by Disertori, Pinson, and Spencer, Comm. Math. Phys. 232:83–124 (2002). At last a recent work by Magnen, Rivasseau, and the author, Markov Process and Related Fields 9:261–278 (2003) is summarized: it gives a toy modeldescribing the lowest order approximation of Anderson model and it is proved that, for d=2, its density of states is given by the semicircle distribution. A short discussion of its extension to d≥3 follows.

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  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160

    Jean Bellissard

  2. Institut Universitaire de France, France

    Jean Bellissard

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Bellissard, J. Random Matrix Theory and the Anderson Model. Journal of Statistical Physics 116, 739–754 (2004). https://doi.org/10.1023/B:JOSS.0000037246.61440.6c

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