Scaling Phenomena in a Unitary Model of Directed Propagating Waves with   Applications to One-Dimensional Electrons in a Time-Varying Potential
  We study a 2D lattice model of forward-directed waves in which the integrated intensity for classical waves (or probability for quantum mechanical particles) is conserved. The model describes the time evolution of 1D quantum particle in a time-varying potential and also applies to propagation of electromagnetic waves in two dimensions within the parabolic approximation. We present a closed form solution for propagation in a uniform system. Motivated by recent studies of non-unitary directed models for localized 2D electrons tunneling in a magnetic field, we then address related theoretical questions of how the interference pattern between constrained-forward paths in this unitary model is affected by the addition of phases corresponding to such a magnetic field. The behavior is found to depend sensitively on the value of $\Phi/\Phi_{0}$, where $\Phi$ is flux per plaquette and $\Phi_{0}$ is the unit of flux quantum. For $\Phi/\Phi_{0} = p/q$ we find the amplitude to be more collimated the larger is the value of $q$. We next consider propagation in a random forward scattering media. In particular the scaling properties associated with the transverse width $x$ of the wave, as function of its distance $t$ from point source, are addressed. We find the moments of $x$ to scale with $t$ in a very different way from what is known for either off-lattice unitary or on-lattice non-unitary systems. The scaling of the moments of the probability $[P^{n}(x,t)]$ (or intensity) at a point $(x=0,t)$ is found to be consistent with a simple behavior $[P^{n}(0,t)] \sim t^{-\frac{n}{2}}$. Implications to the behavior of one-dimensional lattice quantum particles in a dynamically fluctuating 
Dinko Cule and Yonathan Shapir