The Shortest Path Across the Mesoscopic System

We study distribution functions (DF) of mesoscopic hopping conductance numerically by searching for the shortest path. We have found that the distributions obtained by choosing randomly the chemical potentials (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 DFs obtained for one-dimensional systems were found to be quite close to the independent predictions of V.I. Mel'nikov, A.A. Abrikosov and P. Lee et al. For D=2, the DFs both for narrow system and thin film look similar (and close to the 1D case).The distribution function for the conductance of the square sample is nearly Gaussian as predicted by both Altshuler et al and Serota et al.