## Anderson localization for radial tree-like random quantum graphs (2008)

Citations: | 10 - 0 self |

### BibTeX

@TECHREPORT{Hislop08andersonlocalization,

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

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

institution = {},

year = {2008}

}

### OpenURL

### 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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Citation Context ...em on rooted radial trees to an effective half-line problem with certain singularities at the vertices. We present a generalized version of this symmetry reduction in Appendix A for completeness (cf. =-=[SoS02]-=- for the standard case). Transfer matrix methods can now be used to describe solutions to the generalized eigenvalue problem on the effective half-line. We conclude by computing the spectrum of the pe... |

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Citation Context ...e nontrivial dependence of z entering in m1 and the transfer matrix Tz(ω1) which makes the quantum graph problem different from spectral averaging methods considered for other models before (see e.g. =-=[GM03]-=-) where usually only real entries are considered. We can now prove the main result of this section: Theorem E.4. The spectral measure ρ = ρω of H = H(ω) on the line-like graph L = L(ω) splits into two... |

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