We establish a spectral correspondence between random Schrödinger operators and deterministic convolution operators on wreath products, generalizing previous results that relate Lamplighter groups to Schrödinger operators with Bernoulli potentials. Using this correspondence in both directions, we obtain an elementary criterion for the absolute continuity of convolutions on wreath products, Lifschitz tail estimates for Schrödinger operators on Cayley graphs of polynomial growth, and an exact formula for the second moment of the Green function, expressed in terms of the wreath product with an Abelian group of lamps.