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.

ASJC Scopus subject areas

Engineering (all)
Physics and Astronomy (all)

Access to Document

10.1080/17455030802398132

Cite this

@article{b048fe5c6e60482da9367439de41a8d0,

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.",

author = "Hislop, {Peter D.} and Olaf Post",

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 G{\"u}nter 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.",

year = "2009",

doi = "10.1080/17455030802398132",

language = "English",

volume = "19",

pages = "216--261",

number = "2",

}

TY - JOUR

T1 - Anderson localization for radial tree-like random quantum graphs

AU - Hislop, Peter D.

AU - Post, Olaf

N1 - 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.

PY - 2009

Y1 - 2009

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.

UR - http://www.scopus.com/inward/record.url?scp=70749160518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70749160518&partnerID=8YFLogxK

U2 - 10.1080/17455030802398132

DO - 10.1080/17455030802398132

M3 - Article

AN - SCOPUS:70749160518

SN - 1745-5030

VL - 19

SP - 216

EP - 261

JO - Waves in Random and Complex Media

JF - Waves in Random and Complex Media

IS - 2

ER -

Anderson localization for radial tree-like random quantum graphs

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this