Decorrelation in Local Statistics for random operators

In this paper we study the local spectral statistics in the localised region of various random operator models, including the d-dimensional the Anderson model and random Schr\"odinger operators. It is already established, in the above models, that at an energy E, in the localised energy region of the spectrum, where the density of states n (E) > 0, the local eigenvalue statistics XE is a Poisson processes with intensity n (E) L, L being the Lebesgue measure on R. The question of independence of XE, X₄^ for distinct energies was partially solved in the literature. We solve it completely for all the models for which the Minami technique works.