Fractional Moment Estimates for Random Unitary Operators

Abstract

We consider unitary analogs of $d-$dimensional Anderson models on $l^2({\mathbb{Z}}^d)$ 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 $U_{\omega }$.