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.
|Number of pages||46|
|Journal||Waves in Random and Complex Media|
|State||Published - 2009|
Bibliographical noteFunding 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)