Exact-enumeration approach to tunneling in disordered systems

The model of Nguyen, Spivak, and Shklovskii (NSS) for coherent tunneling of strongly localized electrons, by impurities with negative local scattering amplitudes, is studied by exact enumeration of all impurity configurations. We consider the overlap between electronic wave functions on sites that are t lattice spacings apart and examine the dependence of their statistical properties on the hopping length and the concentration p of impurities. We obtain series up to t=5 for general p, and t=9 in the symmetric case of p=1/2. In this interval, the extent of scattering perpendicular to the hopping direction is found to spread as 〈x²〉{~}t^2ζ, with {ζ}=0.62±{}0.05. The probability that the final amplitude maintains its original sign decays as exp[-a(p)t] for p{≥}0.1, casting doubt on the ``sign''-phase transition suggested by NSS.