Level statistics of Anderson model of disordered systems: connection to Brownian ensembles

We find that the statistics of levels undergoing the metal-insulator transition in systems with Gaussian disorder and non-interacting electrons behaves in a way similar to that of the single parametric Brownian ensembles (Dyson 1962 J. Math. Phys. 3 1191). The latter appear during a Poisson → Wigner-Dyson transition, driven by a random perturbation. The analogy provides analytical evidence for the single-parameter scaling of the level correlations in disordered systems as well as a tool to obtain them at the critical point for a wide range of disorders