Tunnelling resistance of a one-dimensional random lattice

The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunnelling resistance of the form R0iebi is assumed for a cluster of i adjacent insulating sites. Three different ensembles are considered and compared. In the first ensemble, the number of insulating "atoms" is fixed and distributed in a linear chain; in the second one, there exists a fixed probability p of having an insulator "atom" occupying a site in a linear chain; and finally in the third one, a line is bent into a circle and the probability p is considered. It is observed that in the thermodynamic limit, the average ensemble resistance per site diverges at the critical filling fraction pc = e−b, while the variance of the resistance diverges at the lower filling fraction [Formula: see text]. Computer simulations of large but finite systems, however, exhibit a much weaker divergence of the resistance per site at pc and no divergence of the variance at pc1.