Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph

We prove the almost sure existence of absolutely continuous spectrum at low disorder for the Anderson model on the simplest example of a product of a regular tree with a finite graph. This graph contains loops of unbounded size.Comment: 30 pages, 2 figure

Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs

We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs.Comment: Some clarifications were added in the introduction and an extra appendix was adde

Absolutely continuous spectrum for the Anderson model on trees

This dissertation is an examination of the absolutely continuous spectrum for the Anderson model on different types of trees. The text is divided into four chapter: an introduction, two main chapters and conclusions. In Chapter 2 the existence of purely absolutely continuous spectrum is proven for the Anderson model on a Cayley tree, or Bethe lattice, of degree K. The method used, a geometric one, is based on some properties of the hyperbolic distance. It is a simplified generalization of a result for K=3 given by R. Froese, D. Hasler and W. Spitzer. In Chapter 3 a similar result is proven for a more general tree which has vertices of degrees 2 and 3 alternating in a periodic manner. The lack of symmetry changes the analysis, making it possible to eliminate one of the steps in the proof for the Cayley tree.Science, Faculty ofMathematics, Department ofGraduat