Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel

We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At $d$ dimensional growth for $d>2$ this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform $d$ dimensional growth with $d<2$ one has pure point spectrum in this energy region. At exactly uniform $2$ dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum ($d\leq 2$) to absolutely continuous spectrum ($d\geq 3)$ for random operators of the type $\mathcal{P}_r \Delta_d \mathcal{P}_r+\lambda \mathcal{V}$ on $\mathbb{Z}^d$, where $\mathcal{P}_r$ is an orthogonal radial projection, $\Delta_d$ the discrete adjacency operator (Laplacian) on $\mathbb{Z}^d$ and $\lambda \mathcal{V}$ a random potential.Comment: 38 pages, 1 figure; Introduction reorganized, Corollary 1.3 added and almost sure essential spectrum now characterized (Proposition 1.4

A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schr\"odinger operators we can improve some result by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular we solve a problem posed in their work.Comment: new version, parts rearrange

Scaling diagram for the localization length at a band edge

A weak-coupling scaling diagram for the Lyapunov exponent and the integrated density of states near a band edge of a random Jacobi matrix is obtained. The analysis is based on the use of a Fokker-Planck operator describing the drift-diffusion of the Pr\"ufer phases