Fractional Moment Estimates for Random Unitary Operators

TL;DR

This paper extends fractional moment methods to analyze localization in random unitary operators, showing that under certain conditions, these operators exhibit exponential decay of matrix elements, indicating localization.

Contribution

It adapts the Aizenman-Molchanov fractional moment approach to unitary operators with absolutely continuous phases, establishing localization results.

Findings

Exponential fractional moment estimates for matrix elements

Almost sure localization of the unitary operators

Conditions for small off-diagonal elements to ensure localization

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 $U_\omega$.