Statistics of the Lyapunov exponent in 1D random periodic-on-average systems

By means of Monte Carlo simulations we show that there are two qualitativel y different modes of localization of classical waves in 1D random periodic- on-average systems. States from pass bands and band edges of the underlying band structure demonstrate single parameter scaling with universal behavio r. States from the interior of the band gaps do not have universal behavior and require two parameters to describe their scaling propel-ties. The tran sition between these two types of behavior occurs in an extremely narrow re gion of frequencies. When the degree of disorder exceeds a certain critical value the single parameter scaling is restored for an entire band gap.