We present a statistical mechanics model with a hyperbolic supersymmetry. This model is expected to qualitatively describe properties of random band matrices in N dimensions eg localization and delocalization. The "spins" in this model may be thought of taking values in a Poincare super-disc. In three dimensions we show that this model has a diffusive phase. In one dimension there is only the localized phase. The analysis of this model relies a family of identities coming from SUSY together with estimates of a random walk on a percolation cluster. The surprising relation of this model to linearly reinforced random walk will also be highlighted. No knowledge of SUSY is needed. This is joint work with M. Disertori and M. Zirnbauer.