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These are applied to prove that the local eigenvalue statistics $\\xi^\\omega_{E}$ and $\\xi^\\omega_{E^\\prime}$, 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 $\\xi^\\omega_{E}(I)$ and $\\xi^\\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose L\\'evy measure has finite support. 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