TY - JOUR

T1 - The canopy graph and level statistics for random operators on trees

AU - Aizenman, Michael

AU - Warzel, Simone

N1 - Funding Information:
Acknowledgements It is a pleasure to thank R. Sims and D. Jacobson for stimulating discussions of related topics. We also thank the referee for useful references concerning the appearance of the canopy graph in other studies. Some of the work was done at the Weizmann Institute (MA), at the Department of Physics of Complex Systems, and at University of Erlangen-Nürnberg (SW), Department of Physics. We are grateful for the hospitality enjoyed there. This work was supported in part by the NSF Grant DMS-0602360 and the Deutsche Forschungsgemeinschaft (Wa 1699/1).

PY - 2006/11

Y1 - 2006/11

N2 - For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended 'canopy graph.' For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular - pure point possibly with singular continuous component which is proven to occur in some cases.

AB - For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended 'canopy graph.' For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular - pure point possibly with singular continuous component which is proven to occur in some cases.

KW - Absolutely continuous spectrum

KW - Anderson localization

KW - Canopy graph

KW - Level statistics

KW - Random operators

KW - Singular continuous spectrum

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U2 - 10.1007/s11040-007-9018-3

DO - 10.1007/s11040-007-9018-3

M3 - Article

AN - SCOPUS:34249675427

VL - 9

SP - 291

EP - 333

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

SN - 1385-0172

IS - 4

ER -