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We present simple, physically motivated, examples where small geometric changes on a two-dimensional graph , combined with high disorder, have a significant impact on the spectral and dynamical properties of the random Schr\"odinger operator obtained by adding a random potential to the graph's adjacency operator. Differently from the standard Anderson model, the random potential will be constant along any vertical line, 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. Joint work with Matos and Schenker