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We prove local laws, i.e. optimal concentration estimates for arbitraryproducts of resolvents of a Wigner random matrix with deterministic matrices inbetween. We find that the size of such products heavily depends on whether someof the deterministic matrices are traceless. Our estimates correctly accountfor this dependence and they hold optimally down to the smallest possiblespectral scale.

Optimal multi-resolvent local laws for Wigner matrices