  We consider products of the matrices associated with the Szeg\H{o} recursion
from the theory of orthogonal polynomials on the unit circle and show that
under suitable assumptions, their norms grow exponentially in the number of
factors. In the language of dynamical systems, this result expresses a uniform
hyperbolicity statement. We present two applications of this result. On the one
hand, we identify explicitly the almost sure spectrum of extended CMV matrices
with non-negative random Verblunsky coefficients. On the other hand, we show
that no Ising model in one dimension exhibits a phase transition. Also, in the
case of dynamically generated interaction couplings, we describe a gap labeling
theorem for the Lee-Yang zeros in the thermodynamic limit.
