The authors investigate the asymptotic (low-frequency or long-time) behavior of random one-dimensional systems described by a master equation of the form $\frac{C_{n}d{P}_n}{\mathrm{dt}}=W_{n,n-1\mathrm{}}(P_{n-1\mathrm{}}-P_n)+W_{n,n+1\mathrm{}}(P_{n+1\mathrm{}}-P_n),$ where either the $C_n$ or the $W_{n,n+1\mathrm{}}(=W_{n+1,n})$ are independent positive random variables. This problem can be mapped onto a variety of physical problems, including one-dimensional particle or excitation diffusion with random spatial transfer rates or random trap depths, low-temperature properties of the random one-dimensional Heisenberg ferromagnet, one-dimensional tight-binding fermion systems with correlated diagonal and off-diagonal disorder, and electrical lines of random conductances or capacitances. Replacing $\frac{d{P}_n}{\mathrm{dt}}$ by $\frac{d^2{P}_n}{d{t}^2}$, the above equations also describe a harmonic chain with random force constants or random masses. Both the random $W$ and the random $C$ problem are shown to reduce to the same mathematical problem, and the authors review the derivation and consequences of its exact asymptotic solution for some general classes of probability densities. In particular, they are able to determine the exact $\omega \to 0$ asymptotic behavior of the single-site Green function, and thus the low-energy density of states and (for diffusion-type problems) the long-time behavior of the autocorrelation function. For diffusion-type problems, the investigators further introduce a scaling hypothesis (based on the assumption of the existence of a single characteristic length) for the time dependence of the excitation amplitude at sites other than the initially excited site. This allows the calculation of the low-frequency diffusion constant, and with the help of the Einstein relation, the low-frequency conductivity. The authors apply their results to several physical systems and, in particular, are able to account quantitatively for the complex low-frequency conductivity of the one-dimensional superionic conductor hollandite. The results may also be relevant to quasi-one-dimensional electronic systems for times (or frequencies) such that diffusion is restricted to a single dimension. The expansion of an earlier scaling approach gives additional insight into the physical meaning of the asymptotic solutions. The researchers further discuss effective-medium-type approximations, which, apart from a numerical prefactor, lead to the correct asymptotic dependences. Comparison is made with the results of alternative approaches to this type of problem, specifically with those of the continuous time random walk. The authors exhibit the remarkable satisfaction of the hyperscaling relations for the specific heat and correlation function critical exponents for the random Heisenberg ferromagnet, even though the exponents themselves are not universal. Finally, aspects of the problem which remain unsolved, or which are only partly resolved, are discussed. These include the derivation of asymptotic expansions beyond the leading term, the derivation of rigorous asymptotic expressions for the conductivity, the calculation of fluctuations in the site occupancy probabilities for the diffusion problem, and the treatment of the random one-dimensional antiferromagnet and random one-dimensional $\mathrm{xy}$ model. The investigators close with a short description of an application of this work to electronic conduction in highly anisotropic materials.

DOI:https://doi.org/10.1103/RevModPhys.53.175