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We prove a Wegner estimate for alloy type models merely assuming that thesingle site potential is lower bounded by a characteristic function of a thickset, that is a particular set of positive measure. The proof is based on twoingredients: New unique continuation principles or uncertainty relations forlinear combinations of eigenfunctions of the Laplacian on cubes from [EV20] andestablished proofs for Wegner estimates, e.g.~from [CHK07].  We obtain a Wegner estimate with optimal volume dependence at all energies,and localisation near the minimum of the spectrum, even for some non-stationaryrandom potentials.  We complement the result by showing that a lower bound on the potential bythe characteristic function of a thick set is necessary for a (translationuniform) Wegner estimate to hold. Hence, we have identified a sharp conditionon the size for the support of random potentials that is sufficient andnecessary for the validity of Wegner estimates.

Wegner estimate and localisation for alloy type operators with minimal support assumptions on the single site potential