Localization in ensemble of sparse random matrices

We study a transition from the localized to the delocalized phase in a novel sparse-random-matrices (SRM) ensemble. Within the scope of a supersymmetric approach it can be described by an order-parameter function. We attribute a clear physical meaning to this function providing a link between our approach and other methods. Equations for a ‘‘mobility edge and inverse participation ratio’’ are derived. Some analogies between the SRM model and mean-field theory of spin glass as well as possible applications to the theory of Anderson localization are discussed.