Decorrelation estimates for random Schrödinger operators with non rank one perturbations

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We prove decorrelation estimates for generalized lattice Anderson models on Zᵈ constructed with finite-rank perturbations in the spirit of Klopp klopp. These are applied to prove that the local eigenvalue statistics ξωₑ and ξωₑ′, associated with two energies E and E' satisfying |E - E'| > 4d, are independent. That is, if I,J are two bounded intervals, the random variables ξωₑ(I) and ξωₑ'(J), are independent and distributed according to a compound Poisson distribution whose Lévy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation.