Localization for transversally periodic random potentials on binary   trees
  We consider a random Schr\"odinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, $Q_r$, and a random transversally periodic potential, $\kappa Q_t$, with coupling constant $\kappa$. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large $\kappa$. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing $\kappa$. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder. 
Richard Froese, Darrick Lee, Christian Sadel, Wolfgang Spitzer, and   G\"unter Stolz
math-ph math.MP