Resonant delocalization for random Schr\"odinger operators on tree graphs

04/2011; DOI: 10.4171/JEMS/389
Source: arXiv

ABSTRACT We analyse the spectral phase diagram of Schr\"odinger operators $ T +\lambda
V$ 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 $(\lambda \ll 1)$ 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

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    ABSTRACT: We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.
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    ABSTRACT: We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.
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