Barry Simon
Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle
(278K, pdf)

ABSTRACT.  For suitable classes of random Verblunsky coefficients, including 
independent, identically distributed, rotationally invariant ones, we prove that if 
\[ 
\mathbb{E} \biggl( \int\frac{d\theta}{2\pi} \biggl|\biggl( \frac{\mathcal{C} + e^{i\theta}}{\mathcal{C} -e^{i\theta}} 
\biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 |k-\ell|} 
\] 
for some $\kappa_1 >0$ and $p<1$, then for suitable $C_2$ and $\kappa_2 >0$, 
\[ 
\mathbb{E} \bigl( \sup_n |(\mathcal{C}^n)_{k\ell}|\bigr) \leq C_2 
e^{-\kappa_2 |k-\ell|} 
\] 
Here $\mathcal{C}$ is the CMV matrix.