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

Journal of the European Mathematical Society (Impact Factor: 1.7). 04/2011; 15(4). DOI: 10.4171/JEMS/389
Source: arXiv


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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    • "It is generally conjectured that the Anderson model shows a phase transition from localization to delocalization. However, the existence of delocalized eigenvectors and regimes of absolutely continuous spectrum has been established rigorously only on trees [5] [8] [27] [37]. It is a frequently discussed question if this phase transition can also be seen in the local spectral statistics on regular graphs that approximate trees. "
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    ABSTRACT: For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results about the Anderson model on the infinite tree we consider random Schr\"odinger operators on finite regular graphs. We study local spectral statistics: We analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore we develop a new approach based on direct comparison of eigenvectors.
    Preview · Article · Jun 2014 · Annales Henri Poincare
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    • "Our motivation to study resonant delocalization as a mechanism for the formation of bands of extended states was in part motivated by recent results on random Schrödinger operators on tree graphs [1] [2]. The mechanism plays there a role even in regimes of very low density of states, and it is of interest to understand its role in other systems with rapid growth of volume reached by n steps. "
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    ABSTRACT: In the presence of disorder Schr\"odinger operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. Such resonant delocalization is considered here in the context of the random Schr\"odinger operator on the complete graph. The operator exhibits many local quasi modes mixed through through a single channel. As was noted and studied from a number of perspective before through most of its spectrum it exhibits localization. However we find that in addition under appropriate conditions the spectrum includes bands of states which are delocalized in the $\ell^1$-though not in $\ell^2$-sense. In such energy regimes the spectrum resembles that of the \v{S}eba process, which is defined here. In analyzing the operator's spectrum we present and apply some generally useful properties of the scaling limits of random functions in the Pick class. The results are in agreement with a heuristic condition which is stated here for the emergence of resonant delocalization in terms of the tunneling amplitude among quasi-modes.
    Full-text · Article · May 2014 · Annales Henri Poincare

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