Classical Hopping Conduction in Random One-Dimensional Systems: Nonuniversal Limit Theorems and Quasilocalization Effects

The $L\to \infty$ asymptotic properties of ${\rho }_L\mathrm{}\left(g\right)\mathrm{}$, the probability distribution of the classical hopping conductivity $g_L$ corresponding to random one-dimensional systems of length $L$, are determined. These properties are nonuniversal, and become anomalous if the probability density $\rho \mathrm{}\left(w\right)\mathrm{}$ of the random near-neighbor hopping rates is such that $\int 0^{\infty }\mathrm{dw}\rho \mathrm{}\left(w\right)\mathrm{}{w}^{-1\mathrm{}}$ does not exist. The associated quasilocalization effects are discussed and their experimental observability is speculated upon.