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

New unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes from \cite{EgidiV-20} are used to prove a Wegner estimate for alloy type models along the lines of previously known proofs, e.~g.~\cite{CombesHK-07}. Due to the strong input we merely have to assume that the single site potential is lower bounded by a characteristic function of a set of positive measure. We obtain a Wegner estimate with optimal volume dependence at all energies, and localisation near the minimum of the spectrum, even for some non-stationary random potentials.