Spectral and Dynamical Contrast on Highly Correlated Anderson-Type Models

We study spectral and dynamical properties of random Schrödinger operators H_Vert=-A_𝔾_Vert+V_ω and H_Diag=-A_𝔾_Diag+V_ω on certain two-dimensional graphs 𝔾_Vert and 𝔾_Diag . Differently from the standard Anderson model, the random potentials are not independent but, instead, are constant along any vertical line, i.e V_ω(n)=ω (n_1) , for n=(n_1,n_2) . In particular, the potentials studied here exhibit long range correlations. We present examples where geometric changes to the underlying graph, combined with high disorder, have a significant impact on the spectral and dynamical properties of the operators, leading to contrasting behaviors for the “diagonal” and “vertical” models. Moreover, the “vertical” model exhibits a sharp phase transition within its (purely) absolutely continuous spectrum. This is captured by the notions of transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon (J Funct Anal 43:1-31, 1981).