Phase transition in a model with non-compact symmetry on Bethe lattice and the replica limit

We solve the O(n,1) nonlinear vector model on the Bethe lattice and show that it exhibits a transition from ordered to disordered state for . If the replica limit is taken carefully, the model is shown to reduce to the corresponding supersymmetric model. The latter was introduced by Zirnbauer as a toy model for the Anderson localization transition. We argue thus that the non-compact replica models describe correctly the Anderson transition features. This should be contrasted to their failure in the case of the level correlation problem.