Anderson localization for radial tree-like random quantum graphs

Peter D. Hislop; Olaf Post

doi:10.1080/17455030802398132

Anderson localization for radial tree-like random quantum graphs

Peter D. Hislop, Olaf Post

Mathematics

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.

Original language	English
Pages (from-to)	216-261
Number of pages	46
Journal	Waves in Random and Complex Media
Volume	19
Issue number	2
DOIs	https://doi.org/10.1080/17455030802398132
State	Published - 2009

Bibliographical note

Funding Information:
PDH thanks Simone Warzel for discussions on random quantum graphs. OP thanks Peter Kuchment for the invitation at TAMU and for useful suggestions concerning the transfer matrix (cf. Section B.1). Additionally, OP would also like to thank Günter Stolz for the invitation to UAB and general remarks on localization. PDH was partially supported by NSF grant DMS-0503784 and OP partially supported by DFG grant Po 1034/1-1.

Funding

PDH thanks Simone Warzel for discussions on random quantum graphs. OP thanks Peter Kuchment for the invitation at TAMU and for useful suggestions concerning the transfer matrix (cf. Section B.1). Additionally, OP would also like to thank G\u00FCnter Stolz for the invitation to UAB and general remarks on localization. PDH was partially supported by NSF grant DMS-0503784 and OP partially supported by DFG grant Po 1034/1-1.

Funders	Funder number
Deutsche Forschungsgemeinschaft	Po 1034/1-1
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-0503784

ASJC Scopus subject areas

General Engineering
General Physics and Astronomy

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10.1080/17455030802398132

Cite this

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title = "Anderson localization for radial tree-like random quantum graphs",

abstract = "We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.",

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N2 - We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.

AB - We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.

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Anderson localization for radial tree-like random quantum graphs

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