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 Schrödinger 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 $X_E,X_{E^{\mathrm{\prime }}}$ for distinct energies was partially solved in the literature. We solve it completely for all the models for which the Minami technique works.