We present simple, physically motivated, examples where small geometric changes on a two-dimensional graph $mathbb{G}$, combined with high disorder, have a significant impact on the spectral and dynamical properties of the random Schrodinger operator $-A_{mathbb{G}}+V_{omega}$ obtained by adding a random potential to the graphs adjacency operator. Differently from the standard Anderson model, the random potential will be constant along any vertical line, i.e $V_{omega}(n)=omega(n_1)$, for $n=(n_1,n_2)in mathbb{Z}^2$, hence the models exhibit long range correlations. Moreover, one of the models presented here is a natural example where the transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon, coexist and allow us to capture a sharp phase transition present in the model.
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