Geodesic
Resonant delocalization for random Schrödinger operators on tree graphs
Journal of the European Mathematical Society, Tome 15 (2013) no. 4, pp. 1167-1222.

Voir la notice de l'article provenant de la source EMS Press

We analyse the spectral phase diagram of Schrödinger operators T+λV on regular tree graphs, with T the graph adjacency operator and V a random potential given by iid random variables. The main result is a criterion for the emergence of absolutely continuous (ac) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials ac spectrum appears at arbitrarily weak disorder (λ≪1) in an energy regime which extends beyond the spectrum of T. Incorporating considerations of the Green function's large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations' ‘free energy function’, the regime of pure ac spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.
DOI : 10.4171/jems/389
Classification : 82-XX, 47-XX, 00-XX
Keywords: Anderson localization, absolutely continuous spectrum, mobility edge, Cayley tree 
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     author = {Michael Aizenman and Simone Warzel},
     title = {Resonant delocalization for random {Schr\"odinger} operators on tree graphs},
     journal = {Journal of the European Mathematical Society},
     pages = {1167--1222},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2013},
     doi = {10.4171/jems/389},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/389/}
}
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Michael Aizenman; Simone Warzel. Resonant delocalization for random Schrödinger operators on tree graphs. Journal of the European Mathematical Society, Tome 15 (2013) no. 4, pp. 1167-1222. doi : 10.4171/jems/389. http://geodesic.mathdoc.fr/articles/10.4171/jems/389/

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