Fractional moment estimates for random unitary operators

We consider unitary analogs of d-dimensional Anderson models on l(2)(Z(d)) defined by the product U(omega)=D(omega)S where S is a deterministic unitary and D(omega) is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of U(omega)(U(omega) - z)(-1), provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for U(omega).