Alain Joye
Fractional Moment Estimates for Random Unitary Operators
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ABSTRACT. 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 $U_\omega$.