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# One-dimensional Anderson Localization: Devil's staircase of Statistical Anomalies

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### Abstract

The statistics of wavefunctions in the one-dimensional (1d) Anderson model of localization is considered. It is shown that at any energy that corresponds to a rational filling factor f=p/q there is a statistical anomaly which is seen in expansion of the generating function (GF) to the order (q-2) in the disorder parameter. We study in detail the principle anomaly at $$f=1/2$$ that appears in the leading order. The transfer-matrix equation of the Fokker-Planck type with a two-dimensional internal space is derived for GF. It is shown that the zero-mode variant of this equation is integrable and a solution for the generating function is found in the thermodynamic limit.

### Author and article information

###### Journal
12 June 2008
0806.2118
10.1063/1.3149496

AIP Conference Proceedings, vol.1134, p.31-35(2009)
4 pages RevTex, 1 picture
cond-mat.dis-nn cond-mat.mes-hall