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 $X_E$ is a Poisson processes with intensity $n(E) \mathcal{L}$, $\mathcal{L}$ being the Lebesgue measure on $\mathbb{R}$. The question of independence of $X_E, X_{E^\prime}$ for distinct energies was partially solved in the literature. We solve it completely for all the models for which the Minami technique works.