Shortest path across a mesoscopic system

We study distribution functions (DF's) of mesoscopic hopping conductance numerically by searching for the shortest path and the results are compared with analytical predictions. We have found that distributions obtained by choosing the chemical potentials randomly (for a fixed impurity configuration), which corresponds to a typical experimental situation, coincide with those obtained when both impurity configuration and chemical potential is chosen randomly, in agreement with the ergodicity hypothesis. The DF's obtained for one-dimensional (1D) systems were found to be quite close to the independent predictions of Mel'nikov and Raikh and Ruzin. For D=2, the DF's both for a narrow system and a thin film look similar (and close to the 1D case), which means that the short 2D still lies in the narrow regime defined by Raikh and Ruzin. The distribution function for the conductance of the square sample is nearly Gaussian as predicted by both Altshuler and Serota Our results also hint that the puncture nature of 2D systems seems to be featured by the position of DF peak and the long tail might show the preference of conductance fluctuation.